Next Number in a Sequence Lesson - III a5 = 7 an = 6 + a(nï¿½2) There is no guarantee that this is the "right" answer, since the "right" answer is the one that your teacher wants. However, this formula does "work", so it should be acceptable to say that the next term is: The next (nineth) term is 6 + a7 = 6 + 13 = 19. Note, however, that the sequence might be simpler even than that. Start listing the first few primes, and see what kind of sequence you get. Compare this result with the values of the above listing. Find the missing term: 2, 8, 27, 85, 260, ____, 2365 I checked, and this didn't appear to be a polynomial sequence. It didn't appear at first to be a recursive sequence, but recursives can be messy and hard to figure out.

But if one is particularly patient and clever, and plays around with this a bit, one might come up with: a1 = 2 a2 = 2ï¿½3 + 2 = 8 a3 = 8ï¿½3 + 3 = 27 a4 = 27ï¿½3 + 4 = 85 a5 = 85ï¿½3 + 5 = 260 The pattern appears to be "take the last term, multiply by three, and then add the number for the sequence term you're on". Continuing, we get: a6 = 260ï¿½3 + 6 = 786 a7 = 786ï¿½3 + 7 = 2365 So the pattern gives us the seventh term we were supposed to get, which confirms that the sixth term should probably be 786. Find the next number: 2, 3, 4, 6, 6, 9, 8, ___ For this one, I noticed that 4 is twice 2, and 6 is twice 3.

Then 6 (the second one) is thrice 2 and 9 is thrice 3. Continuing, 8 is four times 2, so the next number should be four times 3, or 12. You can also view this as equivalent fractions: 2/3, 4/6, 6/9, 8/12, etc. Either way, the next number is 12. By the way, "fiddling around" is very often the best method for finding solutions. Don't be afraid to grab some scratch paper and just try stuff. There's nothing "wrong" with you if an answer isn't immediately obvious to you. Just keep trying. Prime Numbers - Facts, Examples, & Table Of All Up To 1,000 Math > Numbers Prime Numbers A prime number can be divided, without a remainder, only by itself and by 1. For example, 17 can be divided only by 17 and by 1. Some facts: The only even prime number is 2. All other even numbers can be divided by 2.

If the sum of a number's digits is a multiple of 3, that number can be divided by 3. No prime number greater than 5 ends in a 5. Any number greater than 5 that ends in a 5 can be divided by 5. Zero and 1 are not considered prime numbers. Except for 0 and 1, a number is either a prime number or a composite number. A composite number is defined as any number, greater than 1, that is not prime. To prove whether a number is a prime number, first try dividing it by 2, and see if you get a whole number. If you do, it can't be a prime number. If you don't get a whole number, next try dividing it by prime numbers: 3, 5, 7, 11 (9 is divisible by 3) and so on, always dividing by a prime number (see table below). Here is a table of all prime numbers up to 1,000: Important Concepts and Formulas - Numbers -3, -2, -1, 0, 1, 2, 3 ... d.

Rational Numbers Rational numbers can be expressed as @@\dfrac{a}{b}@@ where a and b are integers and @@b \ne 0@@ Examples: @@\dfrac{11}{2}@@, @@\dfrac{4}{2}@@, @@0@@, @@\dfrac{-8}{11}@@ etc. All integers, fractions and terminating or recurring decimals are rational numbers. e. Irrational Numbers Any number which is not a rational number is an irrational number. In other words, an irrational number is a number which cannot be expressed as @@\dfrac{a}{b}@@ where a and b are integers. For instance, numbers whose decimals do not terminate and do not repeat cannot be written as a fraction and hence they are irrational numbers.

Example : @@\pi@@, @@\sqrt{2}@@, @@(3 + \sqrt{5})@@, @@4 \sqrt{3}@@ (meaning @@4 \times \sqrt{3}@@), @@\sqrt[3]{6}@@ etc Please note that the value of @@\pi@@ = 3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510 58209 74944 59230 78164 06286 20899 86280 34825 34211 70679... We cannot @@\pi@@ as a simple fraction (The fraction 22/7 = 3.14.... is just an approximate value of @@\pi@@) f. Real Numbers Real numbers include counting numbers, whole numbers, integers, rational numbers and irrational numbers. g.

Surds Let @@a@@ be any rational number and @@n@@ be any positive integer such that @@\sqrt[n]{a}@@ is irrational. Then @@\sqrt[n]{a}@@ is a surd. Example : @@\sqrt{3} @@, @@\sqrt[6]{10} @@, @@4 \sqrt{3}@@ etc Please note that numbers like @@\sqrt{9}@@, @@\sqrt[3]{27}@@ etc are not surds because they are not irrational numbers Every surd is an irrational number. But every irrational number is not a surd. (eg : @@\pi@@ , @@e@@ etc are not surds though they are irrational numbers.) Addition Rules for Even and Odd Numbers 1. Sum of any number of even numbers is always even 2. Sum of even number of odd numbers is always even 3. Sum of odd number of odd numbers is always odd Subtraction Rules for Even and Odd Numbers 1. Difference of two even numbers is always even 2. Difference of two odd numbers is always even Multiplication Rules for Even and Odd Numbers 1. Product of even numbers is always even 2. Product of odd numbers is always odd 3.

If there is at least one even number multiplied by any number of odd numbers, the product is always even 3. Divisibility Divisible By One whole number is divisible by another if the remainder we get after the division is zero. Examples A number is divisible by 2 if the last digit is even. i.e., if the last digit is 0 or 2 or 4 or 6 or 8 Example 1: Check if 64 is divisible by 2. The last digit of 64 is 4 (even). Hence 64 is divisible by 2 Example 2: Check if 69 is divisible by 2. The last digit of 69 is 9 (not even). Hence 69 is not divisible by 2 A number is divisible by 3 if the sum of the digits is divisible by 3 Please note that we can apply this rule to the answer again and again if needed. Example 1: Check if 387 is divisible by 3. 3 + 8 + 7 = 18. 18 is divisible by 3.

Hence 387 is divisible by 3 Example 2: Check if 421 is divisible by 3. 4 + 2 + 1 = 7. 7 is not divisible by 3. Hence 421 is not divisible by 3 A number is divisible by 4 if the number formed by the last two digits is divisible by 4. Example 1: Check if 416 is divisible by 4. Number formed by the last two digits = 16. 16 is divisible by 4. Hence 416 is divisible by 4 Example 2: Check if 481 is divisible by 4. Number formed by the last two digits = 81. 81 is not divisible by 4. Hence 481 is not divisible by 4 A number is divisible by 5 if the last digit is either 0 or 5. Example 1: Check if 305 is divisible by 5. Last digit is 5. Hence 305 is divisible by 5. Example 2: Check if 420 is divisible by 5. Last digit is 0.

Hence 420 is divisible by 5. Example 3: Check if 312 is divisible by 5. Last digit is 2. A number is divisible by 6 if it is divisible by both 2 and 3. Example 1: Check if 546 is divisible by 6. 546 is divisible by 2. 546 is also divisible by 3. (Refer divisibility rule of 2 and 3 ) Hence 546 is divisible by 6 Example 2: Check if 633 is divisible by 6. 633 is not divisible by 2 though it is divisible by 3. Hence 633 is not divisible by 6 Example 3: Check if 635 is divisible by 6. 635 is not divisible by 2. 635 is also not divisible by 3. Hence 635 is not divisible by 6 Example 4: Check if 428 is divisible by 6. 428 is divisible by 2 but it is not divisible by 3.

Hence 428 is not divisible by 6 To find out if a number is divisible by 7, double the last digit and subtract it from the number formed by the remaining digits. Repeat this process until we get at a smaller number whose divisibility we know. If this smaller number is 0 or divisible by 7, the original number is also divisible by 7. Example 1: Check if 349 is divisible by 7. Given number = 349 34 - (9 × 2) = 34 - 18 = 16 16 is not divisible by 7. Hence 349 is not divisible by 7 Example 2: Check if 364 is divisible by 7. Given number = 364 36 - (4 × 2) = 36 - 8 = 28 28 is divisible by 7. Hence 364 is also divisible by 7 Example 3: Check if 3374 is divisible by 7.

Given number = 3374 337 - (4 × 2) = 337 - 8 = 329 32 - (9 × 2) = 32 - 18 = 14 14 is divisible by 7. Hence 329 is also divisible by 7. Hence 3374 is also divisible by 7. A number is divisible by 8 if the number formed by the last three digits is divisible by 8. Example 1: Check if 7624 is divisible by 8. The number formed by the last three digits of 7624 = 624. 624 is divisible by 8. Hence 7624 is also divisible by 8. Example 2: Check if 129437464 is divisible by 8. The number formed by the last three digits of 129437464 = 464. 464 is divisible by 8. Hence 129437464 is also divisible by 8.

Example 3: Check if 737460 is divisible by 8. The number formed by the last three digits of 737460 = 460. 460 is not divisible by 8. Hence 737460 is also not divisible by 8. A number is divisible by 9 if the sum of its digits is divisible by 9. (Please note that we can apply this rule to the answer again and again if we need) Example 1: Check if 367821 is divisible by 9 3 + 6 + 7 + 8 + 2 + 1 = 27 27 is divisible by 9. Hence 367821 is also divisible by 9. Example 2: Check if 47128 is divisible by 9 4 + 7 + 1 + 2 + 8 = 22 22 is not divisible by 9. Hence 47128 is not divisible by 9.

Example 3: Check if 4975291989 is divisible by 9 4 + 9+ 7 + 5 + 2 + 9 + 1 + 9 + 8 + 9= 63 Since 63 is big, we can use the same method to see if it is divisible by 9. 6 + 3 = 9 9 is divisible by 9. Hence 63 is also divisible by 9. Hence 4975291989 is also divisible by 9. A number is divisible by 10 if the last digit is 0. Example 1:Check if 2570 is divisible by 10. Last digit is 0. Hence 2570 is divisible by 10. Example 2: Check if 5462 is divisible by 10. Last digit is not 0. Hence 5462 is not divisible by 10 To find out if a number is divisible by 11, find the sum of the odd numbered digits and the sum of the even numbered digits.

Now subtract the lower number obtained from the bigger number obtained. If the number we get is 0 or divisible by 11, the original number is also divisible by 11. Example 1: Check if 85136 is divisible by 11. 8 + 1 + 6 = 15 We got the difference as 0. Hence 9548 is divisible by 11. A number is divisible by 12 if the number is divisible by both 3 and 4 Example 1: Check if 720 is divisible by 12. 720 is divisible by 3. 720 is also divisible by 4. (Refer divisibility rules of 3 and 4 ) Hence 720 is divisible by 12 Example 2: Check if 916 is divisible by 12. 916 is not divisible by 3 , though 916 is divisible by 4. Hence 916 is not divisible by 12 Example 3: Check if 921 is divisible by 12. 921 is divisible by 3.

But 921 is not divisible by 4. Hence 921 is not divisible by 12 Example 4: Check if 827 is divisible by 12. 827 is not divisible by 3. 827 is also not divisible by 4. Hence 827 is not divisible by 12 To find out if a number is divisible by 13, multiply the last digit by 4 and add it to the number formed by the remaining digits. Repeat this process until we get at a smaller number whose divisibility we know. If this smaller number is divisible by 13, the original number is also divisible by 13. Example 1:Check if 349 is divisible by 13. Given number = 349 34 + (9 × 4) = 34 + 36 = 70 70 is not divisible by 13. Hence 349 is not divisible by 349 Example 2: Check if 572 is divisible by 13.

Given number = 572 57 + (2 × 4) = 57 + 8 = 65 65 is divisible by 13. Hence 572 is also divisible by 13 Example 3: Check if 68172 is divisible by 13. Given number = 68172 6817 + (2 × 4) = 6817 + 8 = 6825 682 + (5 × 4) = 682 + 20 = 702 70 + (2 × 4) = 70 + 8 = 78 78 is divisible by 13. Hence 68172 is also divisible by 13. Example 4: Check if 651 is divisible by 13. Given number = 651 65 + (1 × 4) = 65 + 4 = 69 69 is not divisible by 13.

Hence 651 is not divisible by 13 A number is divisible by 14 if it is divisible by both 2 and 7. Example 1:Check if 238 is divisible by 14 238 is divisible by 2. 238 is also divisible by 7. (Refer divisibility rule of 2 and 7 ) Hence 238 is divisible by 14 Example 2: Check if 336 is divisible by 14 336 is divisible by 2. 336 is also divisible by 7. Hence 336 is divisible by 14 Example 3: Check if 342 is divisible by 14. 342 is divisible by 2. But 342 is not divisible by 7. Hence 342 is not divisible by 12 Example 4: Check if 175 is divisible by 14. 175 is not divisible by 2, though it is divisible by 7. Hence 175 is not divisible by 14 Example 5: Check if 337 is divisible by 14.

337 is neither divisible by 2 nor by 7 Hence 337 is not divisible by 14 A number is divisible by 15 if it is divisible by both 3 and 5. Example 1: Check if 435 is divisible by 15 435 is divisible by 3. 435 is also divisible by 5. (Refer divisibility rule of 3 and 5 ) Hence 435 is divisible by 15 Example 2: Check if 555 is divisible by 15 555 is divisible by 3. 555 is also divisible by 5. Hence 555 is also divisible by 15 Example 3: Check if 483 is divisible by 15. 483 is divisible by 3 But 483 is not divisible by 5. Hence 483 is not divisible by 15 Example 4: Check if 485 is divisible by 15. 485 is not divisible by 3, though it is divisible by 5.

Hence 485 is not divisible by 15 Example 5: Check if 487 is divisible by 15. 487 is not divisible by 3. It is also not divisible by 5 Hence 487 is not divisible by 15 A number is divisible by 16 if the number formed by the last four digits is divisible by 16. Example 1: Check if 5696512 is divisible by 16. The number formed by the last four digits of 5696512 = 6512 6512 is divisible by 16. Hence 5696512 is also divisible by 16. Example 2: Check if 3326976 is divisible by 16. The number formed by the last four digits of 3326976 = 6976 6976 is divisible by 16. Hence 3326976 is also divisible by 16. Example 3: Check if 732374360 is divisible by 16.

The number formed by the last three digits of 732374360 = 4360 4360 is not divisible by 16. Hence 732374360 is also not divisible by 16. To find out if a number is divisible by 17, multiply the last digit by 5 and subtract it from the number formed by the remaining digits. Repeat this process until you arrive at a smaller number whose divisibility you know. If this smaller number is divisible by 17, the original number is also divisible by 17. Example 1: Check if 500327 is divisible by 17.

Given Number = 500327 50032 - (7 × 5 )= 50032 - 35 = 49997 4999 - (7 × 5 ) = 4999 - 35 = 4964 496 - (4 × 5 ) = 496 - 20 = 476 47 - (6 × 5 ) = 47 - 30 = 17 17 is divisible by 17. Hence 500327 is also divisible by 17 Example 2: Check if 521461 is divisible by 17.

Given Number = 521461 52146 - (1 × 5 )= 52146 -5 = 52141 5214 - (1 × 5 ) = 5214 - 5 = 5209 520 - (9 × 5 ) = 520 - 45 = 475 47 - (5 × 5 ) = 47 - 25 = 22 22 is not divisible by 17. Hence 521461 is not divisible by 17 A number is divisible by 18 if it is divisible by both 2 and 9. Example 1: Check if 31104 is divisible by 18. 31104 is divisible by 2. 31104 is also divisible by 9. (Refer divisibility rule of 2 and 9 ) Hence 31104 is divisible by 18 Example 2: Check if 1170 is divisible by 18. 1170 is divisible by 2.

1170 is also divisible by 9. Hence 1170 is divisible by 18 Example 3: Check if 1182 is divisible by 18. 1182 is divisible by 2 But 1182 is not divisible by 9. Hence 1182 is not divisible by 18 Example 4: Check if 1287 is divisible by 18. 1287 is not divisible by 2 though it is divisible by 9. Hence 1287 is not divisible by 18 To find out if a number is divisible by 19, multiply the last digit by 2 and add it to the number formed by the remaining digits. Repeat this process until you arrive at a smaller number whose divisibility you know. If this smaller number is divisible by 19, the original number is also divisible by 19. Example 1: Check if 74689 is divisible by 19.

Given Number = 74689 7468 + (9 × 2 )= 7468 + 18 = 7486 748 + (6 × 2 ) = 748 + 12 = 760 76 + (0 × 2 ) = 76 + 0 = 76 76 is divisible by 19. Hence 74689 is also divisible by 19 Example 2: Check if 71234 is divisible by 19. Given Number = 71234 7123 + (4 × 2 )= 7123 + 8 = 7131 713 + (1 × 2 )= 713 + 2 = 715 71 + (5 × 2 )= 71 + 10 = 81 81 is not divisible by 19.

Hence 71234 is not divisible by 19 Divisibility by 20 A number is divisible by 20 if it is divisible by 10 and the tens digit is even. (There is one more rule to see if a number is divisible by 20 which is given below. A number is divisible by 20 if the number is divisible by both 4 and 5) Example 1: Check if 720 is divisible by 20 720 is divisible by 10. (Refer divisibility rule of 10 ). The tens digit = 2 = even digit. Hence 720 is also divisible by 20 Example 2: Check if 1340 is divisible by 20 1340 is divisible by 10. The tens digit = 4 = even digit. Hence 1340 is divisible by 20 Example 3: Check if 1350 is divisible by 20 1350 is divisible by 10. But the tens digit = 5 = not an even digit.

Hence 1350 is not divisible by 20 Example 4: Check if 1325 is divisible by 20 1325 is not divisible by 10 though the tens digit = 2 = even digit. Hence 1325 is not divisible by 20 4. What are Factors of a Number and how to find it out? Factors of a number If one number is divisible by a second number, the second number is a factor of the first number. The lowest factor of any positive number = 1 The highest factor of any positive number = the number itself. Example The factors of 36 are 1, 2, 3, 4, 6, 9 12, 18, 36 because each of these numbers divides 36 with a remainder of 0 How to find out factors of a number Write down 1 and the number itself (lowest and highest factors).

Check if the given number is divisible by 2 (Reference: Divisibility by 2 rule ) If the number is divisible by 2, write down 2 as the second lowest factor and divide the given number by 2 to get the second highest factor Check for divisibility by 3, 4,5, and so on. till the beginning of the list reaches the end Example 1: Find out the factors of 72 Write down 1 and the number itself (72) as lowest and highest factors. 1 . . . 72 72 is divisible by 2 ( Reference: Divisibility by 2 Rule ). 72 ÷ 2 = 36. Hence 2nd lowest factor = 2 and 2nd highest factor = 36. So we can write as 1, 2 . . . 36, 72 72 is divisible by 3 ( Reference: Divisibility by 3 Rule ). 72 ÷ 3 = 24 . Hence 3rd lowest factor = 3 and 3rd highest factor = 24. So we can write as 1, 2, 3, . . .

24, 36, 72 72 is divisible by 4 ( Reference: Divisibility by 4 Rule ). 72 ÷ 4 = 18. Hence 4th lowest factor = 4 and 4th highest factor = 18. So we can write as 1, 2, 3, 4, . . . 18, 24, 36, 72 72 is divisible by 6 ( Reference: Divisibility by 6 Rule ). 72 ÷ 6 = 12. Hence 5th lowest factor = 6 and 5th highest factor = 12. So we can write as 1, 2, 3, 4, 6, . . . 12, 18, 24, 36, 72 72 is divisible by 8 ( Reference: Divisibility by 8 Rule ). 72 ÷ 8 = 9. Hence 6th lowest factor = 8 and 6th highest factor = 9.

Now our list is complete and the factors of 72 are 1, 2, 3, 4, 6, 8, 9 12, 18, 24, 36, 72 Example 2: Find out the factors of 22 Write down 1 and the number itself (22) as lowest and highest factors 1 . . . 22 22 is divisible by 2 ( Reference: Divisibility by 2 Rule ). 22 ÷ 2 = 11. Hence 2nd lowest factor = 2 and 2nd highest factor = 11. So we can write as 1, 2 . . . 11, 22 22 is not divisible by 10 ( Reference: Divisibility by 10 Rule ). Now our list is complete and the factors of 22 are 1, 2, 11, 22 If a number is divisible by another number, then it is also divisible by all the factors of that number.

Example : 108 is divisible by 36 because 106 ÷ 38 = 3 with remainder of 0. The factors of 36 are 1, 2, 3, 4, 6, 9 12, 18, 36 because each of these numbers divides 36 with a remainder of 0. Hence, 108 is also divisible by each of the numbers 1, 2, 3, 4, 6, 9, 12, 18, 36. 5. What are Prime Numbers and Composite Numbers? Prime Numbers A prime number is a positive integer that is divisible by itself and 1 only. Prime numbers will have exactly two integer factors. Examples: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, etc. Please note the following facts Zero is not a prime number because zero is divisible by more than two numbers. Zero can be divided by 1, 2, 3 etc.

(0 ÷ 1 = 0, 0÷ 2 = 0 ...) One is not a prime number because it does not have two factors. It is divisible by only 1 Composite Numbers Composite numbers are numbers that have more than two factors. A composite number is divisible by at least one number other than 1 and itself. Examples: 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, etc. Please note that zero and 1 are neither prime numbers nor composite numbers.

Every whole number is either prime or composite, with two exceptions 0 and 1 which are neither prime nor composite Prime factorization of a number is the expression of the number as the product of its prime factors Example 1: Prime factorization of 280 can be written as 280 = 2 × 2 × 2 × 5 × 7 = 23 × 5 × 7 and the prime factors of 280 are 2, 5 and 7 Example 2: Prime factorization of 72 can be written as 72 = 2 × 2 × 2 × 3 × 3 = 23 × 32 and the prime factors of 72 are 2 and 3 Repeated Division Method : In order to find out the prime factorization of a number, divide the number repeatedly by the smallest prime number possible(2,3,5,7,11, ...) until the quotient is 1.

Example 1: Find out prime factorization of 280 2 3 1 Hence, prime factorization of 72 can be written as 72 = 2 × 2 × 2 × 3 × 3 = 23 × 32 and the prime factors of 72 are 2 and 3 Important Properties Every whole number greater than 1 can be uniquely expressed as the product of its prime factors. For example, 700 = 22 × 52 × 7 Multiples of a whole number are the products of that number with 1, 2, 3, 4, and so on Example : Multiples of 3 are 3, 6, 9, 12, 15, ... If a number x divides another number y exactly with a remainder of 0, we can say that x is a factor of y and y is a multiple of x For instance, 4 divides 36 exactly with a remainder of 0. Hence 4 is a factor of 36 and 36 is a multiple of 4 Type 1: Fractions with same denominators.

Compare @@\dfrac{3}{5}@@ and @@\dfrac{1}{5}@@ These fractions have same denominator. So just compare the numerators. Bigger the numerator, bigger the number. 3 > 1. Hence @@\dfrac{3}{5} > \dfrac{1}{5}@@ Example 2: Compare @@\dfrac{2}{7}@@ and @@\dfrac{3}{7}@@ and @@\dfrac{8}{7}@@ These fractions have same denominator. So just compare the numerators. Bigger the numerator, bigger the number. 8 > 3 > 2. Hence @@\dfrac{8}{7} > \dfrac{3}{7} > \dfrac{2}{7}@@ Type 2 : Fractions with same numerators. Example 1: Compare @@\dfrac{3}{5}@@ and @@\dfrac{3}{8}@@ These fractions have same numerator. So just compare the denominators. Bigger the denominator, smaller the number. 8 > 5.

Hence @@\dfrac{3}{8} < \dfrac{3}{5}@@ Example 2: Compare @@\dfrac{7}{8}@@ and @@\dfrac{7}{2}@@ and @@\dfrac{7}{5}@@ These fractions have same numerator. So just compare the denominators. Bigger the denominator, smaller the number. 8 > 5 > 2. Hence @@\dfrac{7}{8} < \dfrac{7}{5} < \dfrac{7}{2}@@ Type 3 : Fractions with different numerators and denominators. Example 1: Compare @@\dfrac{3}{5}@@ and @@\dfrac{4}{7}@@ To compare such fractions, find out LCM of the denominators. Here, LCM(5, 7) = 35 Now , convert each of the given fractions into an equivalent fraction with 35 (LCM) as the denominator. The denominator of @@\dfrac{3}{5}@@ is 5. 5 needs to be multiplied with 7 to get 35.

Hence, @@\dfrac{3}{5} = \dfrac{3 \times 7}{5 \times 7} = \dfrac{21}{35}@@ The denominator of @@\dfrac{4}{7}@@ is 7. 7 needs to be multiplied with 5 to get 35. Hence, @@\dfrac{4}{7} = \dfrac{4 \times 5}{7 \times 5} = \dfrac{20}{35}@@ @@\dfrac{21}{35} > \dfrac{20}{35}@@ Hence, @@\dfrac{3}{5} > \dfrac{4}{7}@@ Or Convert the fractions to decimals @@\dfrac{3}{5} = .6@@ @@\dfrac{4}{7} = .5...@@ (Need not find out the complete decimal value; just find out up to what is required for comparison. In this case the first digit itself is sufficient to do the comparison) .6 > .5...

Two numbers are said to be co-prime (also spelled coprime) or relatively prime if they do not have a common factor other than 1. i.e., if their HCF is 1. Example 1: 3, 5 are co-prime numbers (Because HCF of 3 and 5 = 1) Example 2: 14, 15 are co-prime numbers (Because HCF of 14 and 15 = 1) A set of numbers is said to be pairwise co-prime (or pairwise relatively prime) if every two distinct numbers in the set are co-prime Example 1: The numbers 10, 7, 33, 13 are pairwise co-prime, because HCF of any pair of the numbers in this is 1. HCF (10, 7) = HCF (10, 33) = HCF (10, 13) = HCF (7, 33) = HCF (7, 13) = HCF (33, 13) = 1.

Example 2 : The numbers 10, 7, 33, 14 are not pairwise co-prime because HCF(10, 14) = 2 ≠ 1 and HCF(7, 14) = 7 ≠ 1. If a number is divisible by two co-prime numbers, then the number is divisible by their product also. Example Types of Numbers As you can see, the location of the ones digit in the binary representation indicates the numbers of the binary sequence that are to be added together to yield the base 10 number of interest. CARDINAL NUMBERS A cardinal number is a number that defines how many items there are in a group or collection of items. Typically, an entire group of items is referred to as the "set" of items and the items within the set are referred to as the "elements" of the set. For example, the number, group, or set, of players on a baseball team is defined by the cardinal number 9. The set of 200 high school students in the graduating class is defined by the cardinal number 200.

(See ordinal numbers and tag numbers.) CATALAN NUMBERS Catalan numbers are one of many special sequences of numbers that derive from combinatorics problems in recreational mathematics. Combinatorics deals with the selection of elements from a set of elements typically encountered under the topics of probability, combinations, permutations, and sampling. The specific Catalan numbers are 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16,796 and so on deriving from This particular set of numbers derive from several combinatoric problems, one of which is the following. Given "2n" people gathered at a round table. How many person to person, non-crossing, handshakes can be made, i.e., no pairs of arms crossing one another across the table? A few quick sketches of circles with even sets of dots and lines will lead you to the first three answers easily. Two people, one handshake. Four people, two handshakes. Six people, 5 handshakes. With a little patience and perseverance, eight people will lead you to 14 handshakes.

Beyond that, it is probably best to rely on the given expression. CHOICE NUMBERS Choice numbers, more commonly called combination numbers, or simply combinations, are the number of ways that a number of things can be selected, chosen, or grouped. Combinations concern only the grouping of items and not the arrangement of those items. They typically evolve from the question how many combinations of "n" objects are possible using all "n" objects or "r" objects at a time? To find the number of combinations of "n" dissimilar things taken "r" at a time, the formula is: which can be stated as "n" factorial divided by the product of "r" factorial times (n - r) factorial. Example: How many different ways can you combine the letters A, B, C, and D in sets of three? Clearly, namely ABC, ABD, ACD, and BCD. (Note that ACB, BAC, BCA, CAB and CBA are all the same combination just arranged differently. In how many ways can a committee of three people be selected from a group of 12 people?

We have: How many handshakes will take place between six people in a room when they each shakes hands with all the other people in the room one time? Here, Notice that no consideration is given to the order or arrangement of the items but simply the combinations. Another way of viewing combinations is as follows. Consider the number of combinations of 5 letters taken 3 at a time. This produces: Now assume you permute (arrange) the r = 3 letters in each of the 10 combinations in all possible ways. Each group would produce r! permutations. Letting x = 5C3 for the moment, we would therefore have a total of x(r!) different permutations. This total, however, represents all the possible permutations (arrangements) of n things taken r at a time, which is shown under arrangement numbers and defined as nPr. Therefore, Using the committee of 3 out of 12 people example from above, Consider the following: How many different ways can you enter a 4 door car? It is clear that there are 4 different ways of entering the car.

Another way of expressing this is: If we ignore the presence of the front seats for the purpose of this example, how many different ways can you exit the car assuming that you do not exit through the door you entered? Clearly you have 3 choices. This too can be expressed as: Carrying this one step further, how many different ways can you enter the car by one door and exit through another? Entering through door #1 leaves you with 3 other doors to exit through. The same result exists if you enter through either of the other 3 doors. Therefore, the total number of ways of entering and exiting under the specified conditions is: Another example of this type of situation is how many ways can a committee of 4 girls and 3 boys be selected from a class of 10 girls and 8 boys? This results in: See ARRANGEMENT NUMBERS for determining the number of possible arrangements between items. CIRCULAR PRIMES A circular prime number is one that remains a prime number after repeatedly relocating the first digit of the number to the end of the number. For example, 197, 971 and 719 are all prime numbers.

Similarly, 1193, 1931, 9311 and 3119 are all prime numbers. Other numbers that satisfy the definition are 11, 13, 37, 79, 113, 199 and 337. Primes of two or more digits can only contain the digits 1, 3, 7 because, If 0, 2, 4, 5,6, or 8 were part of the number, in the units place, the number would be divisible by 2 or 5. It is thought that there are an infinite number of circular primes but has not yet been proven. COMPLEX NUMBERS Complex numbers are formed by the addition of a real number and an imaginary number, the general form of which is a + bi where i = = the imaginary number and a and b are real numbers. The "a" is said to be the real part of the complex number and b the imaginary part. Probably the easiest number to define after prime numbers. The Fundamental Theorem of Arithmetic states that every positive integer greater than 1 is either a prime number or a composite number.

As we know, a prime number "p" is any positive number the only divisors of which are 1 and p (or -1 and -p). Thus, by definition, any number that is not a prime number must be a composite number. A composite number is any number having 3 or more factors/divisors and is the result of multiplying prime numbers together. Most of the positive integers are the product of smaller prime numbers. Examples: 4, 6, 8, 10, 12, 14, 15, 16, 18, 20, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, etc., are all composite numbers, each being divisible by lower prime numbers. Every number divisible by 2, the only even prime, is composite.

Every composite number can be broken down to a single unique set of prime factors and their exponents. Examples: 210 = 2 x 3 x 5 x 7; 495 = 32 x 51 x 111 or 4500 = 2 x 2 x 3 x 3 x 5 x 5 x 5 = 22 x 32 x 53. This is the one and only possible factorization of the number 210. If a positive number N is evenly divisible by any prime number less than , the number N is composite. Wilson's Theorem states that for every prime number "p", [(p + 1)! + 1] is evenly divisible by "p". The converse was shown to also be true in that every integer "N" that evenly divides [(N + 1)! + 1] is prime. Combining these leads to the famous general theorem that a necessary and sufficient condition that an integer "N" be prime is that "N" evenly divide [(n + 1)! + 1]. Conversely, if "N" does not divide [(N + 1)!

+ 1], "N" is composite. Unfortunately, the practical use of this method is minimal due to the large numbers encountered with high N's. CONGRUENT NUMBERS A number N is said to be congruent if there are two integers, x and y, that result in the expressions x2 + Ny2 and x2 - Ny2 being perfect squares. The smallest known congruent number is 5 which satisfies 412 + 5(122) = 492 and 412 - 5(122) = 312. The use of the square of a negative number results in another solution of 22 + 5(12) = 32 and 22 - 5(12) = (-1)2. While there are many congruent numbers, finding them is an arduous task. The expressions x2 + Ny2 and x2 - Ny2 are often useful in solving many problems in recreational mathematics.

COUNTING NUMBERS The counting numbers are the familiar set of whole numbers, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11,....., that we see and use every day. (The 0 is sometimes included.) The set of counting numbers is often referred to as the natural numbers. CUBIC NUMBERS or CUBES A cubic number is the third power of a number as in a x a x a = a3. Those familiar with the evolution of the squares from adding successive odd numbers might not be too surprised to discover how the cubes evolve from summing odd numbers also. Clearly, the nth cube is simply n3. Cubes can be derived in other ways also: The cube of any integer, "n", is the sum of the series of odd numbers beginning with (n2 - n + 1) and ending with (n2 + n - 1).

Example: For n = 6, (n2 - n + 1) = 31 and 31 + 33 + 35 + 37 + 39 + 41 = 216 = 63. The last line of numbers are the perfect cubes. The first ten cubes are 1, 8, 27, 64, 125, 216, 343, 512, 729 and 1000. The sum of the first n cubes starting with 1, is 13 + 23 + 33 +........+ n3. which, quite surprisingly, is the square of the nth triangular number, defined by Tn = n(n+1)/2. The sum of the cubes of the first n odd numbers is 2n4 - n2 = n2(2n2 - 1). The sum of the cubes of the first n even numbers is 2n4 + 4n3 + 2n2 = 2n2(n + 1)2.

The sum of the first n cubes, 13 + 23 + 33 + 43 +........+ n3 is equal to the square of the sum of the first n integers. Thus, 13 + 23 + 33 + 43 +........+ n3 = (1 + 2 + 3 + 4 +........+ n)2. The cube of any integer is the difference of the squares of two other integers. Every cube is either a multiple of 9 or right next to one. A perfect cube can end in any of the digits 0 through 9. Three digit numbers that are the sum of the cubes of their digits: 153, 370, 371, 407. The smallest number that is the sum of 2 cubes in two different ways. 1729 = 13 + 123 = 103 + 93. What are the dimensions of two cubes with integral sides that have their combined volume equal to the combined length of their edges. What are the dimensions of the cubes? x = 2 and y = 4.

The sum of any two cubes can never be a cube. The sum of a series of three or more cubes can equal a cube. Examples: 113 + 123 + 133 + 143 = 203 11343 + 11353 + .........21333 = 16,8303 The cube root of a number N is the number "a" which, when multiplied by itself twice, results in the number N or N = axaxa. There is no formula for extracting the cube root of a number. It can be obtained by means of a long division method or a simple estimation method. The sum of "n" terms of an arithmetic progression with the first term equal to the sum of the first "n" natural numbers and a common difference of "n" is n3. First, a method for approximating the cube root of a number to several decimal places which is usually sufficient for everyday use. Estimation Method 1--Make an estimate of the cube root of N = n lying between successive integers a and b.

2--Compute A = N - a3 and B = b3 - N 2--n = a + (bA)/[bA + aB] Example: Find the cube root of 146 1--With 146 lying between 125 and 216, let a = 5 and b = 6. 2--A = 21 and B = 70 3--n = 5 + (6 x 21)/[6 x 21 + 5 x 70] = 5 + 126/476 = 5 + .264 = 5.2647 4--The cube root of 146 is 5.2656 A more exact method for determining integer cube roots.

Note that each cube of the numbers 1 through 10 ends in a different digit: n3....1......8....27....64...125...216...343...512...729...1000 Also note that the last digit is the cube root for all cases except 2, 3, 7 and 8. A quick review of these exceptions leads to the fact that these four digits are the difference between 10 and the cube root, i.e., 8 = 10 - 2, 7 = 10 - 3, 3 = 10 - 7 and 2 = 10 - 8. How can this information be used to determine the cube root of a number? Given the cube of a number between 1 and 100, say 300,763. The last digit tells us that the last digit of the cube root is 10 - 3 = 7. Eliminating the last 3 digits of the cube leaves the number 300. The number 300 lies between the cubes of 6 and 7 in our listing above.

The first digit of our cube root will be the lowest of these two numbers, in this case 6. Therefore, the cube root of 30,763 becomes 67. The last digit of 4 is the last digit of the cube root. The number 592 lies between 512 and 729, the cubes of 8 and 9. Therefore, the cube root of 592,704 becomes 84. Of course, this is only useful if you know ahead of time that the cube is a perfect cube, i.e., having an integral cube root. CYCLIC NUMBERS A cyclic number is a number of "n" digits that when multiplied by 1, 2, 3,...n, results in the same digits but in a different order.

For example, the number 142,857 is a cyclic number since 142,857 x 2 = 285,714, 142,857 x 3 = 428,571, 142,857 x 4 = 571,428, and so on. It is not known just how many cyclic numbers exist. DECIMAL NUMBERS Decimal numbers are numbers expressed through the decimal, or base 10, number system where each digit represents a multiple of some power of 10. The term applies primarily to numbers that have fractional parts so indicated by a decimal point. A number less than 1 is called a decimal fraction, e.g., .673. A mixed decimal is one consisting of an integer and a decimal fraction, e.g., 37.937. . Rational numbers can be expressed in the form of a fraction, 1/2, or as a decimal, .50, 1/8, or as a decimal, .125.

From experience, we know that a fraction expressed in decimal form will either terminate without a remainder such as 3/8 = 0.375 or 7/8 = 0.875, repeat the same digit endlessly such as 1/3 = .3333333..... or 2/3 = .6666666....., repeat a series of different digits repeatedly such as 1/27 = .037037037... or 1/7 = .142857142857...., or repeat a series of digits after some non repeating digits such as 1/12 = .0833333..... All prime denominators produce repeating decimals. Fractions with the same denominator often produce decimals with the same period and period length but with the digits starting with a different number in the period.

For instance, 1/7 = .142857142857142857...., 2/7 = .285714285714285714...., 3/7 = .428571428571428571...., 4/7 = .571428571428571428...., 5/7 = .714285714285714285...., and 6/7 = .857142857142857142.... Other denominators produce two or more repeating periods in different orders. Investigate those with prime denominators of 11, 13, 17, 19, etc. and then 9, 12, 14, 16, 18, etc.and see what results. What do you notice? Another interesting property of repeating decimals of even period length is illustrated by the following.

Take the decimal equivalent of 2/7 = .285714285714...... The repeating period is 285714. Split the period into two groups of three digits and add them together. The result is 999. Do the same with any other repeating decimal period and the result will always be a series of nines. Take 1/17 = .0588235294117647........ Adding 05882352 and 94117647 gives you 99999999. All repeating decimals, regardless of the period and length, are rational numbers. This simply means that it can be expressed as the quotient of two integers. A question that frequently arises is how to convert a repeating decimal, which we know to be rational, back to a fraction. Rational decimal fractions may be converted to fractions as follows: Given the decimal number N = 0.078078078... Multiply N by 1000 or 1000N = 78.078078078...

Subtracting N = .078078078... .................................999N = 78 making N = 78/999 = 26/333. Given the decimal number N = .076923076923... Muliply N by 1,000,000 or 1,000,000N = 76,923.0769230769230... Subtracting N = .0769231769230... .........................................999,999N = 76,923 making N = 76,923/999,999 = 1/13 An easier way to derive the fraction is to simply place the repeating digits over the same number of 9's. For example, the repeating decimal of .729729729729 converts to the fraction of 729/999 = 27/37.

Another trick where zeros are involved is to place the repeating digits over the same number of 9's with as many zeros following the 9's as there are zeros in the repeating decimal. For example, .00757575 leads to 075/9900 = 1/132. DEFICIENT NUMBERS Deficient numbers are part of the family of numbers that are either deficient, perfect, or abundant. Deficient numbers, dN, are numbers where the sum of its aliquot parts (proper divisors), sa(N), is less than the number itself sa(N) < N. (In the language of the Greek mathematicians, the divisors of a number N were defined as any whole numbers smaller than N that, when divided into N, produced whole numbers. The factors/divisors of a number N, less the number itself, are referred to as the aliquot parts, or aliquot divisors, of the number.) Equivalently, N is also deficient if the sum, s(N), of all its divisors is less than 2N.

It can be readily seen that using the aliquot parts summation, sa(16) = 1+2+4+8 = 14 < N = 16 while using all of the divisors, s(16) = 1+2+4+8+16 = 31 < 2N = 32, making the number 16 deficient under either definition. 1,2,3,4,5,7,8,9,10,11,13,14,15,16,17,19,21,22,23 are all deficient. A prime number or any power of a prime number is deficient. The divisors of a pefect or deficient number is deficient. DIGITAL ROOT The digital root of a number is the single digit that results from the continuous summation of the digits of the number and the numbers resulting from each summation. For example, consider the number 7935. The summation of its digits is 24. The summation of 2 and 4 is 6, the digital root of 7935.

Digital roots are used to check addition and multiplication by means of a method called casting out nines. For example, check the summation of 378 and 942. The DR of 378 is 9, 3+7+8=18, 1+8=9.. The DR of 942 is 6, 9+4+2=15, 1+5=6. Adding 9 and 6 produces 15, The DR of 15 is 6, 1+5=6. The summation of 378 and 942 is 1320. The DR of 1320 is 6. With the two final DR's are equal, the addition is correct. EGYPTIAN FRACTIONS Egyptian fractions are the reciprocals of the positive integers where the numerator is always one. They are often referred to as unit fractions. They were used exclusively by the Egyptians to represent all forms of fractions. The two fractions that they used that did not have a unit fraction was 2/3 and 3/4.

The only other fractions that they seemed to have a strong interest in were those of the form 2/n where n was any positive odd number. The Rhind papyrus contains a list of unit fractions representing a series of 2/n for odd n's from 5 to 501. It is unclear as to why they found these 2/n fractions so important. These and other 2/n fractions may be derived from 2/n = 1/[n(n+1)/2] + 1/[n+1)/2]. In the year 1202, Leonardo Fibonacci proved that any ordinary fraction could be expressed as the sum of a series of unit fractions in an infinite number of ways. He used the then named greedy method for deriving basic unit fraction expansions. He described the greedy method in his Liber Abaci as simply subtracting the largest unit fraction less than the given non unit fraction and repeating the process until only unit fractions remained. It was later shown that the greedy method, when applied to any fraction m/n, results in a series of no more than "m" unit fractions. An example will best illustrate the process. Reduce the fraction 13/17 to a sum of unit fractions.

Dividing the fraction yields .7647. Of the unit fractions 1/2, 1/3, 1/4, 1/5, etc., 1/2 is the largest that is smaller than 13/17 so we compute 13/17 - 1/2 = 9/34 making 13/17 = 1/2 + 9/34. Repeating with 9/34, we have 9/34 - 1/4 = 2/136 = 1/68 making 13/17 = 1/2 + 1/4 + 1/68. Alternatively, divide the numerator into the denominator and use the next highest integer as the new denominator. 17/13 = 1.307 making 2 the denominator of the fraction to be subtracted from 13/17. Reduce the fraction 23/37 to a sum of unit fractions.

23/37 - 1/2 = 9/74 - 1/16 = 70/1184 - 1/17 = 6/20,128 - 1/3355 = 1/33,764,720 making 23/37 = 1/2 + 1/16 + 1/17 + 1/3355 + 1/33,764,720. There are many methods or algorithms that derive the unit fractions for any fraction m/n. Much more information regarding unit fractions can be found at Unit Fractions Search at Egyptian Fractions at ~eppstein/numth/egypt/ Unit Fraction from Math World at Algorithms for Egyptian Fractions at ~eppstein/numth/egypt/force.html Creating Unit Fractions at The unit fractions derived by means of the method shown, or any other method, can be further broken down into other unit fractions by means of the identity 1/a = 1/(a+1) + 1/a(a+1), also known to Fibonacci.

For example, 1/2 = 1/(2+1) + 1/2(2+1) = 1/3 + 1/6. Further still, 1/3 = 1/(3+1) + 1/3(3+1) = 1/4 + 1/12 and 1/6 = 1/(6+1) + 1/6(6+1) = 1/7 + 1/42 yielding 1/2 = 1/4 + 1/7 + 1/12 + 1/42. While the number of unit fractions derivable for any given fraction is therefore infinite, there is apparently no known procedure for deriving a series with the least number of unit fractions or the smallest largest denominator. . EQUABLE TRIANGLES A shape is called equable if its area equals its perimeter. There are exactly five equable Heronian triangles: the ones with side lengths (5,12,13), (6,8,10), (6,25,29), (7,15,20), and (9,10,17).

EQUIVALENT NUMBERS Equivalent numbers are numbers where the aliquot parts (proper divisors other than the number itself) are identical. For instance, 159, 559 and 703 are equivalent numbers since their aliquot parts sum to 57. f(159) = 1, 3, 53 and 159 where 1 + 3 + 53 = 57. f(559) = 1, 13, 43 and 559 where 1 + 13 + 43 = 57. f(703) = 1, 19, 37 and 703 where 1 + 19 + 37 = 57. EVEN NUMBERS Probably the easiest number to define, an even number is any number that is evenly divisible by 2. The nth even number is given by Ne = 2n. The sum of the set of "n" consecutive even numbers beginning with 2 is given by Se = n(n + 1).

The sum of the set of m consecutive even numbers starting with n1 and ending with n2 is given by Se(n1-n2) = n2^2 - n1^2 + (n1 + n2) or (n1 + n2)(1 + n1 - n2). The sum of the squares of the even numbers starting with 2^2 is given by Se^2 = (4n^3 + 6n^2 + 2n)/3. An even number multiplied by any number, or raised to any power, results in another even number. The number 2 is the only even prime number. Prime numbers (2,3,5,7,11,13,...) Prime Numbers Is 2 a prime number? What is prime number? Prime number is a positive natural number that has only two positive natural number divisors - one and itself. The opposite of prime numbers are composite numbers. A composite number is a positive nutural number that has at least one positive divisor other than one or itself. The number 1 is not a prime number by definition - it has only one divisor.

The number 0 is not a prime number - it is not a positive number and has infinite number of divisors. The number 15 has divisors of 1,3,5,15 because: 15/1=15 So 15 is not a prime number. The number 13 has only two divisors of 1,13. 13/1=13 So 13 is a prime number. Prime numbers list List of prime numbers up to 100: 2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97,... Is 0 a prime number? The number 0 is not a prime number. Zero is not a positive number and has infinite number of divisors. Is 1 a prime number? The number 1 is not a prime number by definition. One is has one divisor - itself. Is 2 a prime number? The number 2 is a prime number.

Two has 2 natural number divisors - 1 and 2: 2 / 1 = 2

Question: 2, 3, 7, 11, and 19 are all examples of what type of number?
Answer: Prime