MU-NLPC/Calc-aqua_rat huggingface dataset:
A line has a slope of 3/4 and intersects the point T (-12, -39). At which point does this line intersect the x-axis?
Choose the most appropriate option
A) (40,0)
B) (30,0)
C) (0,40)
D) (40,30)
E) (0,30)

Solution: Assume that the equation of the line is y=mx+c, where m and c are the slope and y-intercept.   You are also given that the line crosses the point (-12,-39), this means that this point will also lie on the line above.   Thus you get -39=m*(-12)+c , with m=3/4 as the slope is given to be 3/4.   After substituting the above values, you get c = -30.   Thus the equation of the line is y=0.75*x-30 and the point where it will intersect the x-axis will be with y coordinate = 0.   Put y=0 in the above equation of the line and you will get, x=40.   Thus, the point T of intersection is (40,0).   (40,0) is the correct answer.
Answer: A
At a dinner party, 6 people are to be seated around a circular table. Two seating arrangement are considered different only when the positions of the people are different relative to each other. What is the total number of different possible seating arrangements for the group? Choose the most appropriate option.
A) 5 B) 10 C) 120 D) 320 E) 120

Solution: Has to be C.   Simple formula: There are (n - 1)! ways to arrange n distinct objects in a circle   hence (6-1)= 
(6-1) = 5
5!= 120
Answer: C
Bob has a jar of fruit, and he ate 1/8 of the fruit for lunch. If Bob ate 3/7 of the remaining fruit for dinner, what fraction of the fruit was left after dinner?
Choose the correct choice from the following options:
A) 1/4  B) 1/3  C) 1/2  D) 2/5  E) 3/7

Solution: Let x be the amount of fruit at the beginning.   After lunch, the remaining fruit was (7/8)x.   After dinner, the remaining fruit was (4/7)(7/8)x = (1/2)x.   The answer is 1/2.
Answer: C
What is the sum between the place values of two 3's in the numeral 63130 Select the correct option:
A) 3111  B) 300  C) 3029  D) 3030  E) 3003

Solution: Required sum = 3000+30= 
3000+30 = 3_030
3030   Answer is 3030
Answer: D
In 1970 there were 8,902 women stockbrokers in the United States. By 1978 the number had increased to 18,947. Approximately what was the percent increase? Select:
A) 45%	B) 113%	C) 145%	D) 150%	E) 225%

Solution: The percent increase is (18947-8902)/8902=10045/8902=1.13   So the approximate answer is 113%
Answer: B
If 12 men do a work in 80 days, in how many days will 16 men do it?  Choose the most appropriate option:
A) 22
B) 87
C) 77
D) 60
E) 85

Solution: 12 * 80 = 16 * x   x = 60 days   Answer: 60
Answer: D
A train 125 m long passes a man, running at 7 km/hr in the same direction in which the train is going, in 10 seconds. The speed of the train is?  Choose the correct choice from the following choices:  A) 11  B) 50  C) 52  D) 65  E) 22

Solution: Speed of the train relative to man = (125/10) m/sec = (25/2) m/sec. [(25/2) * (18/5)] km/hr   = 45 km/hr. Let the speed of the train be x km/hr. Then, relative speed = (x - 7) km/hr. x - 7 = 45 ==> x   = 52 km/hr.   Answer:52
Answer: C
A bag contains 6 black and 8 white balls. One ball is drawn at random. What is the probability that the ball drawn is white?  Choose the correct choice from the following options:  A) 3/4  B) 4/7  C) 1/7  D) 1/8  E) 4/3

Solution: Let number of balls = (6 + 8)= 
(6 + 8) = 14
14.   Number of white balls= 8.   P (drawing a white ball) =8/14=4/7.   Option 4/7.
Answer: B
A train moves with a speed of 108 kmph. Its speed in metres per second is
Choose the correct option: A) 10.8 B) 18 C) 25 D) 30 E) None

Solution: Sol.   108 kmph = [108 x 5/18] m/sec = 30 m/sec.   Answer 30
Answer: D
A bag contains 3 white marbles and 3 black marbles. If each of 3 girls and 3 boys randomly selects and keeps a marble, what is the probability that all of the girls select the same colored marble?
Choose the correct choice from the following answers:
A) 1/35
B) 1/10
C) 1/15
D) 1/20
E) 1/25

Solution: first, total ways to select for all boys and girls, i.e 6!/(3!*3!) = 6*5*4*3*2*1/3*2*1*3*2*1=20   then there are one two way girls can have all same colors, either white or black.   The number of ways in which 3 girls can select 3 white balls = 3C3 = 1   The number of ways in which 3 girls can select 3 black balls = 3C3 = 1   Therefore, total favorable outcomes/total outcomes = 2/20= 1/10   1/10
Answer: B
A store reduced the price of all items in the store by 12% on the first day and by another 10% on the second day. The price of items on the second day was what percent of the price before the first reduction took place?	Pick
A) 80.0
B) 79.2
C) 81.0
D) 81.1
E) 81.9

Solution: consider price of the all items as $100   after a initial reduction of 12% price becomes = 0.88 *100= 
0.88 *100 = 88
$ 88   after the final reduction of 10% price becomes= 0.9* 88= 
0.9* 88 = 79.2
$ 79.2   price of all items on second day is 79.2% of price on first day   correct answer option 79.2
Answer: B
Claire has a total of 96 pets consisting of gerbils and hamsters only. One-quarter of the gerbils are male, and one-third of the hamsters are male. If there are 25 males altogether, how many gerbils does Claire have?  Choose the correct choice from the following:  A) 84  B) 50  C) 54  D) 57  E) 60

Solution: G+H =96 ...1 ; G/4 +H/3 =25 ....2 OR 3G +4H =25*12= 
25*12 = 300
300   G=96-H OR 3(96-H) +4H =300 H = 300-288= 
300-288 = 12
12 THEN G= 96-12= 
96-12 = 84
84   84
Answer: A
A trader sells 40 metres of cloth for Rs.8200 at a profit of Rs.20 per metre of cloth. How much profit will the trder earn on 40 metres of cloth?
Choose the correct choice from the following answers:	A) Rs.800	B) Rs.1500	C) Rs.1000	D) Rs.1200	E) None of these

Solution: Explanation:   SP of 1 metre cloth = 8200/40 = Rs.205.   CP of 1 metre cloth = Rs.205 – 20 = Rs.185   CP on 40 metres = 185 x 40 = Rs.7400   Profit earned on 40 metres cloth = Rs.8200 – Rs.7400 = Rs.800.   Rs.800nswer: Option Rs.800
Answer: A
A cricket bat is sold for $850, making a profit of $255. The profit percentage would be
Choose one
A) 24%  B) 25%  C) 30%  D) 36%  E) 43%

Solution: 255/(850 - 255) = 255/595 = 51/119=43% .   Answer: 43%.
Answer: E
If the volume and surface area of a sphere are numerically the same, then its radius is:?
Choose the correct answer:
A) 2	B) 4	C) 1	D) 5	E) 3

Solution: 4/3 πr(power3) = 4 π r(power2)   r = 3   Answer is 3.
Answer: E
Eleven bags are bought for Rs.1000 and sold at 10 for Rs.1100. What is the gain or loss in percentage?
Choose the most appropriate option:
A) 10%  B) 21%  C) 25%  D) 20%  E) None of these

Solution: Explanation:   As selling price is Rs.1100 for 10 bags, each bag is sold at Rs.110.   Hence, the profit is Rs.100 after selling 10 bags plus the selling price of 1 bag = 100 + 110 = Rs.210.   %profit = 210/1000 *100= 
210/1000 *100 = 21
21%   ANSWER 21%
Answer: B
Selling an kite for Rs. 30, a shop keeper gains 20%. During a clearance sale, the shopkeeper allows a discount of 10% on the marked price. His gain percent during the sale is?  Choose the correct answer.
A) 8%
B) 10%
C) 11%
D) 15%
E) 20%

Solution: Explanation:   Marked price = Rs. 30   C.P. = 100/120 * 30 = Rs. 25   Sale price = 90% of Rs. 30 = Rs. 27   Required gain % = 0.2/25 * 100 = 8%.   8%nswer: 8%
Answer: A
Before a salary increase, the weekly salary of a worker for 44 hours in a week was as much as the worker is paid now for 40 hours of work in a week. What is the percent increase in the worker's salary per hour?
Choose the correct choice from the following answers
A) 4	B) 6	C) 8	D) 10	E) 12

Solution: Let x be the original salary per hour.   Let y be the salary per hour after the increase.   40y=44x   y=(44/40)= 
(44/40) = 11/10 = around 1.1
1.1x   The salary increased 10%.   The answer is 10.
Answer: D
The length of a room is 5.5 m and width is 3.75 m. What is the cost of paying the floor by slabs at the rate of Rs.800 per sq. metre. Choose the most appropriate option:	A) 12000	B) 19500	C) 18000	D) 16500	E) 20000

Solution: AREA = 5.5 * 3.75 SQ MT   COST FOR 1 SQ M = 800   TOT COST = 5.5 * 3.75 * 800 = 5.5 = 3000 = 16500   ANSWER 16500
Answer: D
There are 7 couples. If they will sit 14 chairs in a row such that each couple sits side by side, how many possible cases are there?	Choose one: A) 120 B) 240 C) 1,200 D) 2,460 E) 654120

Solution: Tie the couples. Then we have 5 persons that have to be arranged in 4 places. 4! ways.   Now the couples can change position with each other. 2! ways.   7! * (2!)^7=5040 *128= 
5040 *128 = 645_120
645120   Answer is 654120.
Answer: E
A manufacturer of a certain type of screw rejects any screw whose length is less than 2.5 — 0.03 centimeters or greater than 2.56 - 0.03 centimeters. If k represents the length of a screw, in centimeters, which of the following inequalities specifies all the lengths of screws that are acceptable? Choose one.
A) |k + 0.03| > 2.5	B) |k — 0.03| <= 2.5	C) |k — 2.5| > 0.03	D) |k — 2.5| >= 0.06	E) |k — 2.5| <= 0.03

Solution: So, let's go through this step by step:   rejects any screw whose length is less than 2.5 — 0.03 centimeters or greater than 2.5 + 0.03 centimeters.   In other words, any screw that is less than: 2.50 - 0.03= 
2.50 - 0.03 = 2.47
2.47 or greater than   2.56 - 0.03= 
  2.56 - 0.03 = 2.53
2.53 will be rejected.   If k represents the length of a screw   In other words,Kis an acceptable screw that must fall within the acceptable range of 2.47 to 2.53, So:   2.47 ≤ K ≤ 2.53   You can rule out answers with < or > as opposed to ≤ or ≥ because the length cannot be LESS than 2.47 or GREATER than 2.53. In other words, 2.47 and 2.53 are acceptable lengths.   Let's look at (|k — 2.5| <= 0.03):   |k — 2.5| <= 0.03   For the positive case: k - 2.5 ≤ 0.03 ===>k ≤ 2.53   For the negative case: -(k - 2.5) ≤ 0.03 ===> -k +2.5 ≤ 0.03 ===> - k ≤ -2.47 ===>k ≥ 2.47   2.47 ≤ k ≤ 2.53   (|k — 2.5| <= 0.03)
Answer: E
The length of a rectangle is halved, while its breadth is tripled. Watis the % change in area?
Choose one:
A) 43% B) 47% C) 50% D) 65% E) 68%

Solution: Let original length = x and original breadth = y.   Original area = xy.   New length = x .   2   New breadth = 3y.   New area = x x 3y = 3 xy.   2 2   Increase % = 1 xy x 1 x 100 % = 50%.   2 xy   50%
Answer: C
Concentrated plums juice comes inside a cylinder tube with a radius of 2.5 inches and a height of 15 inches. The tubes are packed into wooden boxes, each with dimensions of 11 inches by 10 inches by 31 inches. How many tubes of concentrated plums juice, at the most, can fit into 3 wooden boxes?
Choose the correct option.
A) 24.
B) 28.
C) 36.
D) 42.
E) 48.

Solution: Concentrated plums juice comes inside a cylinder tube since height of a tube is 15 inches , the tubes can fit only in one way .   now, diameter of each tube = 5 inches   therefore, 4*2 can be put in each wooden box   in 3 boxes  3*4*2= 
 3*4*2 = 24
24=24.
Answer: A
If Ajay completes a certain work in 8 days and Vijay can complete the same work in 24 days, find the time in which the work will be completed if both Ajay and Vijay work together. Choices
A) 12 days  B) 16 days  C) 6 days  D) 10 days  E) None of these

Solution: Explanation:   On working together Ajay and Vijay can complete the work in (8*24)/(8 + 24)   = 192/32= 
192/32 = 6
6 days   ANSWER: 6 days
Answer: C
To be eligible for retirement benefits at the Sieman Corporation, the sum of an employee's age, in years, and the number of years employment must be at least 62. If a, where a <62, is a certain employee's age when hired and b is the minimum number of years of employment required for the employee to be eligible for retirement benefits, which of the following equations represents the relationship between a and b?
Choose the correct choice from the following answers:
A) ab=62
B) a+2b=62
C) 2a+2b=62
D) 2a+b=62
E) a+b=62

Solution: We are told that when a certain employee was hired he was a years old and that the he needed minimum b years to be eligible for benefits.   Now,after b years of employment, when the employee will be be eligible for benefits, the employee will be a+b years old and will have b years of employment, thus{age after b years}+{employment years}=(a+b)+b=a+2b=62   Answer: a+2b=62
Answer: B
A disease has infected an ant colony that consists of "workers" and "soldiers": 30% of the soldiers and 14% of the "workers" were infected. If that colony is inhabited by 7000 ants, out of which 1700 are infected, how many "workers" inhabit the colony?  Options.
A) 1500	B) 2500	C) 2750	D) 3000	E) 3250

Solution: From two question we get wo equations:   (1) 0,3s + 0,14w = 1700   (2) s + w = 7000   From (2) we get w=7000-s   In (1) 0,3s + 980-0,14s= 1700   0,16s=720   72000:16= 4500 =s   In (2) 4500 + w = 7000   w= 2500   ANSWER:2500
Answer: B
A bag contains 7 red, 5 blue and 4 green balls. If 2 ballsare picked at random, what is the probability that both are red?
Choose the most appropriate option
A) 2/15 B) 2/21 C) 7/40 D) 3/29 E) 4/27

Solution: P(Both are red),   =77/402167/402=77/402167/402   =21/120= 7/40   7/40
Answer: C
Rs.385 were divided among X,Y,Z in such a way that X had Rs.20 more than Y and Z had Rs 15 more than X . How much was Y’s share? Options:
A) Rs.130 B) Rs.145 C) Rs.154 D) Rs.160 E) Rs.164

Solution: Let Y gets Rs x. Then We can say X gets Rs (x + 20 ) and Y gets Rs ( x + 35) .   x + 20 + x + x + 35 = 385   3x = 330   x = 110 .   R’s share = Rs ( 110 + 35 ) = Rs.145   Rs.145
Answer: B
Susan drove an average speed of 30 miles per hour for the first 60 miles of a tripthen at a average speed of 60 miles/hr for the remaining 30 miles of the trip if she made no stops during the trip what was susan's avg speed in miles/hr for the entire trip	Choose the correct answer:	A) 35	B) 36	C) 45	D) 50	E) 55

Solution: Avg. speed = total distance / total time   Total distance = 90 miles   Total time = 60 / 30 + 30/60 = 5/2   Avg. speed = 36   Answer - 36
Answer: B
M = {-6, -5, -4, -3, -2}
T = {-3, -2, -1, 0, 1, 2, 3, 4}
If an integer is to be randomly selected from set M above and an integer is to be randomly selected from set T above, what is the probability that the product of the two integers will be negative?	Choose the correct choice from the following answers:
A) 0	B) 1/3	C) 2/5	D) 1/2	E) 3/5

Solution: We will have a negative product only if 1, 2, 3, or 4 are selected from set T.   P(negative product) = 4/8 = 1/2   The answer is 1/2.
Answer: D
A man travelled for 13 hours. He covered the first half of the distance at 20 kmph and remaining half of the distance at 25 kmph. Find the distance travelled by the man?  Choose the correct choice from the following:
A) 168 km  B) 864 km  C) 200 km  D) 240 km  E) 460 km

Solution: Let the distance travelled be x km.   Total time = (x/2)/20 + (x/2)/25 = 13 => x/40 + x/50   = 13 => (5x + 4x)/200 = 13 => x   = 200 km   Answer: 200 km
Answer: C
a and b are both positive, even integers. The lowest common multiple of a and b can be written as (a·b)/n, where n is an integer. Which of the following may be false?
Choose the correct answer:
A) n is a factor of both a and b  B) a·b is even  C) (a·b)/n is even  D) (a·b)/n < a·b  E) n is a multiple of 4

Solution: A. n is a factor of both a and b : True since LCM is an integer and (a−b)/n is an integer.   B. a·b is even : True since both a and b are even, a-b is also even.   C. (a·b)/n is even : Hold   D. (a·b)/n < a·b : True since n is an integer   E. n is a multiple of 4 : Hold   Plugin values for a and b and E is the only choice that can't be true always.   Answer is n is a multiple of 4
Answer: E
When a certain tree was first planted, it was 4 feet tall, and the height of the tree increased by a constant amount each year for the next 6 years. At the end of the 6th year, the tree was 1/3 taller than it was at the end of the 4th year. By how many feet did the height of the tree increase each year?
Choose the correct option.
A) 3/10 B) 2 C) 1/2 D) 2/3 E) 6/5

Solution: Say, the tree grows by x feet every year.   Then, 4 + 6x = (1+1/3)(4+4x)   or, x = 2   Answer 2
Answer: B
Find the minimum value of n such that 50! is perfectly divisible by 2520^n.  Answers.
A) 0
B) 1
C) 2
D) 3
E) 4

Solution: 50!/2520^n   2520 -> 2^3 * 3^2 * 5^1 * 7^1   Here 7 is the largest prime factor...   So in order to find the minimum value of "n", it is enough to find the minimum power of "7"... nd for maximum value of "n", find max power of 7...   For max. value of n, find   50/7^1 + 50/7^2 = 7 + 1= 
7 + 1 = 8
8 [quotient only]   Max. value of n which is perfectly divisible by 2520^n is (8)   Min. value is 1   Max value : 8   Min Value : 1   ANSWER:1
Answer: B
Select the non termination decimal among the following. Choose the correct choice from the following:	A) 7/3	B) 3/4	C) 4/5	D) 7/8	E) 9/10

Solution: A. Non terminating decimal means that 7/3 = 2.333333 and so on.
Answer: A
2y - 3x = 2xy and x ≠ 0. If x and y are integers, which of the following could equal y?  Choose one: A) 2 B) 1 C) 0 D) 4 E) - 2

Solution: Plug in the answer choices in the equation from the question stem.   A) y = 2 >>> 4-x = 4x >>> No value of x will satisfy this, not even 0. POE   B) y = 1 >>> 2 - x = 2x >>> Same, POE   C) y = 0 >>> -x = 0 >>> x can not equal 0   E) y = -2 >>> -4 - 3x = -4x >>> Holds true for x = 4, no need to check E. This is the answer.   Answer - 2
Answer: E
A women buys some shoestrips and then 4 times of that she buys packet pins and then 8 times of shoestrips she buys handkerchiefs Finally she has a bill of Rs3.24. She pays for each article as many paise as there are articles(of that particular item).Now whats the number of handkerchiefs?
Choose the correct option:
A) 15  B) 16  C) 17  D) 18  E) 19

Solution: 16   ANSWER:16
Answer: B
A watch was sold at a loss of 20%. If it was sold for Rs.520more, there would have been a gain of 6%. What is the cost price? Select the correct option:
A) Rs.1500 B) Rs.2000 C) Rs.3000 D) Rs.4200 E) Rs.2500

Solution: 80%   106%   --------   26% ---- 520   100% ---- ? => Rs.2000   Answer:Rs.2000
Answer: B
Sid began driving from home on a trip averaging 30 miles per hour. How many miles per hour must Carla drive on average to catch up to him in exactly 3 hours if she leaves 30 minutes after Sid?  Choose one
A) 35
B) 55
C) 39
D) 40
E) 60

Solution: Carla starts 30 minutes later and it takes 3 hr for carla to meet Sid   So Sid total time travelled = 3hr + 30= 
+ 30 = 30
30 * (3 1/2)= 105   so carla need to travle 105 to meet Sid in 3 hrs   speed of carla = 105/3= 
105/3 = 35
35 miles per hour   35nswer is 35
Answer: A
What will come in place of the x in the following Number series? 73205, 6655, 605, 55, x
Choose the correct option:  A) 32564  B) 73205  C) 62343  D) 54756  E) 78564

Solution: (73205)   5 x 11 = 55, 55 x 11 = 605, 605 x 11 = 6655, 6655 x 11 = 73205
Answer: B
Rahim bought 65 books for Rs.1150 from one shop and 50 books for Rs.920 from another. What is the average price he paid per book ?  Choose the correct choice from the following answers
A) Rs.17
B) Rs.18
C) Rs.12
D) Rs.11
E) Rs.10

Solution: Average price per book   = (1150 + 920) / (65 + 50) = 2070 / 115   = Rs.18   Answer:Rs.18
Answer: B
If a no. when divided by 44, gives 432 as quotient and 0 as remainder. What will be the remainder when dividing thesame no. by 31	Select the correct option
A) 2  B) 4  C) 5  D) 7  E) 8

Solution: P ÷ 44 = 432   => P = 432 * 44= 
432 * 44 = 19_008
19008   P / 31= 19008 / 31 = 613, remainder = 5   5
Answer: C
In a office work is distribute between p persons. If 1/8 members are absent then work increased for each person is?	Choose one.
A) 13.28%	B) 14.28%	C) 15.28%	D) 16.28%	E) 17.28%

Solution: let total % of work is 100%   total person=p   1/8 person are absent of total person . so absent person is 1/8p ie p/8.   left person is, p-p/8=7p/8.   p person do the work 100%   1 person do the work 100*p%   7p/8 person do the work (100*p*8)/7p%=114.28%   work increased for each person is=(114.28-100)= 
(114.28-100) = 14.28
14.28%   ANSWER:14.28%
Answer: B
If n is the smallest integer such that 864 times n is the square of an integer, what is the value of n? Select the correct option:
A) 2
B) 3
C) 6
D) 12
E) 24

Solution: 865 can written as = 2 * 2 * 2 * 2*2 * 3 * 3 * 3 --> 2^5 * 3^3 ---(1)   so for 864 * n to be a square of an integer, the integer should have even powers to the prime numbers it composed of.   here 2 already has even power -> So n has to be 3 to make the power of 3 in (1) even   Option 3 is correct
Answer: B
Set M contains number that satisfy the condition that, if integer x is in the set then x + 5 will also be in the set M. If -4 is one value in the set, which of the following values must also be present in the set M? I) -9 II) 1 III) 6  Select the correct option:
A) I only
B) II only
C) I and II only
D) II and III only
E) I, II and III

Solution: Since -4 is in the set, then the set must also include 1, 6, 11, 16, etc...   Depending on whether -4 is the starting element in the set or not, we can have -9...or not.   The answer is II and III only.
Answer: D
In a party attended by 14 persons, each clinch their glass with every other. How many glass clinches?
Choose the correct choice from the following choices:
A) 91
B) 92
C) 93
D) 94
E) 95

Solution: Total no. of person = 11   Total no. of glass clinches = n(n-1)/2   =14*13/2= 
14*13/2 = 91
91   91NSWER:91
Answer: A
A boat can travel with aspeed of 30km/hr in still water. If the speed of the stream is 5km/hr, find the time taken by the boat to go 70km downstream.
Choose the correct option:
A) 1 hr B) 2 hrs C) 3 hrs D) 4 hrs E) 5 hrs

Solution: Speed downstream = (30 + 5) km/hr = 35 km/hr.   Time taken to travel 68 km downstream = 70/35= 
70/35 = 2
2 hrs.   2 hrs
Answer: B
A certain farmer pays $70 per acre per month to rent farmland. How much does the farmer pay per month to rent a rectangular plot of farmland that is 360 feet by 605 feet? (43,560 square feet = 1 acre)  Select the correct option.
A) $5,330	B) $3,360	C) $1,350	D) $360	E) $1050

Solution: Basically the question an error. 1 acre = 43,560 square feet and if it is then the answer is 1050 ($1050)
Answer: E
The amounts of time that three secretaries worked on a special project are in the ratio of 2 to 3 to 5. If they worked a combined total of 90 hours, how many hours did the secretary who worked the longest spend on the project?	Choose the correct choice:
A) 80 B) 70 C) 56 D) 45 E) 14

Solution: 10x = 90   => x = 9   Therefore the secretary who worked the longest spent 9 x 5 = 45 hours on the project   Option (56)
Answer: C
If n is a positive integer and n^2 is divisible by 264, then the largest positive integer that must divide n is	Choose the correct option:
A) 6
B) 12
C) 24
D) 36
E) 48

Solution: The question asks aboutthe largest positive integer that MUST divide n, not COULD divide n. Since the least value of n for which n^2 is a multiple of 72 is 12 then the largest positive integer that MUST divide n is 12.   Complete solution of this question is given above. Please ask if anything remains unclear.   I spent a few hours on this one alone and I'm still not clear. I chose 12 at first, but then changed to 48.   I'm not a native speaker, so here is how I interpreted this question:the largest positive integer that must divide n=the largest positive factor of n. Since n is a variable (i.e. n is moving), so is its largest factor. Please correct if I'm wrong here.   I know that if n = 12, n^2 = 144 = 2 * 72 (satisfy the condition). When n = 12, the largest factor of n is n itself, which is 12. Check: 12 is the largest positive number that must divide 12 --> true   However if n = 48, n^2 = 48 * 48 = 32 * 72 (satisfy the condition too). When n = 48, the largest factor of n is n itself, which is 48. Check: 48 is the largest positive number that must divide 48 --> true   So, I also notice that the keyword isMUST, notCOULD. The question is, why is 48 notMUST divide 48, but instead onlyCOULD divide 48? I'm not clear right here. Why is 12MUST divide 12? What's the difference between them?   Only restriction we have on positive integer n is that n^2 is divisible by 72. The least value of n for which n^2 is divisible by 72 is 12, thus nmustbe divisible by 12 (n is in any case divisible by 12). For all other values of n, for which n^2 is divisible by 72, n will still be divisible by 12. This means that n is always divisible by 12 if n^2 is divisible by 72.   Now, ask yourself: if n=48, is n divisible by 48? No. So, n is not always divisible by 48.   48
Answer: E