# Problem based on divisibility - CMI 2015 -problem 3

## Problem based on divisibility

The problem is based upon the divisibility and prime factorization of a numbers. Also we have to deal with the number divisible by either one prime number or more than one prime numbers.

## Try the problem

A positive integer n is called a magic number if it has the following property: if a and b
are two positive numbers that are not coprime to n then a + b is also not coprime to n.
For example, 2 is a magic number, because sum of any two even numbers is also even.
Which of the following are magic numbers? Write your answers as a sequence of four
letters (Y for Yes and N for No) in correct order.
(i) 129 (ii) 128 (iii) 127 (iv) 100.

I.S.I. Entrance 2015 for B. sc. program at CMI Sub problem 3

Divisibility and Prime factorisation

6 out of 10

Secrets in Inequalities.

## Use some hints

Take the LCM, and point out all the numbers that divides the given number, now select any two of them or any two of the prime factors out of calculated ones.

Now we can say those two prime factors a and b, and then we can easily calculate a+b. now check weather a+b and the number itself has any co prime factor or not other than 1.

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