Question

Let m be a prime number and "a" be a positive integer, if m divides "a square" , then show that m divides"a".

Answer 1

Good question there. Had to look it up myself.

Now, any number can be expressed as a product of primes.

so a is a number.

Let $a = p_1*p_2*p_3.....*p_n$

where $p_1, p_2, ... p_n are all prime factors of a$

now squaring a would mean squaring the right hand side too.

$a^{2} = p_1*p_1*p_2*p_2*p_3*p_3....p_n*p_n$

if m is a prime and it divides a square then it must be one of its prime factors.

Take m as p1 or p2 or anything. Some $p_x$

All the prime factors of a squared are included in a.

So this is the proof. It's quite logical.

If m is one of the prime factors or a square, then it is one of the prime factors of a.