Arithmetic+ 06: UNIZOR.COM  - Math+ &Problems - Arithmetic
Channel: Zor Shekhtman on UNIZOR Education
Date: 2024-03-18

UNIZOR.COM - Creative Mind through Art of Mathematics
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Arithmetic+ 06

Problem A
A positive integer number in decimal notation contains only digits 1 and 0.
The digit 1 occurs 111 times, while the digit 0 occurs an unknown number of times.
Can this number be a square of another integer number?

Hint A: Use the rules of divisibility.

Answer A: No.


Problem B
This problem is based on Problems 04(B,C) of this course Math+ & Problems.
Prove that if the sum of digits of some natural number N is the same as the sum of digits of the number k·N, where k−1 is not divisible by 3, then number N is divisible by 9.

Hint B:
Problem 04(B) stated that a remainder of the division of some natural number by 9 is the same as a remainder of the division by 9 of the sum of this number's digits.
Therefore, both N and k·N have the same remainder if divided by 9.


Problem C
Given a number N with a decimal representation 999...9 that contains k digits 9.
Assume for definitiveness, k is a prime number.
Find a number whose decimal representation contains only digits 1 that is divisible by N.

Answer C: 111...1 should contain 9·k digits 1.

Example C: For N=99 (k=2) the number 111...1 that contains 9·2=18 digits 1 is divisible by N.


Problem D
Consider the number
N=(k+1)·(k+2)·...·(2k−1)·(2k)
How many 2's, depending on k, are in the representation of this number as a product of prime numbers?

Hint D: Notice that
N=(2k)!/(k!)
Then (2k)! can be represented as a product of only odd numbers by only even numbers.

Answer D: N=2k·M
where M is an odd number,
so the number of 2's in the representation of number N as a product of prime numbers is k.