Question

Why does the product of two numbers -having no common factors- that results in a perfect square, make them-the two numbers- perfect squares themselves?

 

Answer 1

states that every positive integer > 1 can be uniquely expressed as a product of primes. These primes may be occur more than once.

 

A positive integer n is a perfect square if and only if each prime that divides n appears an even number of times in this prime factorization.

 

Now suppose a⋅b is a perfect square, and gcd(a,b)=1 .

 

Since a⋅b is a perfect square, each prime p that divides a⋅b appears an even number of times, say 2e , in the prime factorization of a⋅b . Now the number of times p appears in a⋅b equals the number of times it appears in a plus the number of times it appears in b .

 

Since gcd(a,b)=1 , the prime can’t appear in the prime factorization of both a and b .

 

Therefore, p appears 2e times in one of them and zero times in the other. This is true for each prime p that divides a⋅b .

 

For primes p that do not divide a⋅b , p appears zero times in each.

 

We have shown that p appears an even number of times in both a and b , for each prime p . We conclude that both a and b are perfect squares

An extension of this argument yields the following generalization:

 

If a1⋅a2⋅a3⋯an is a perfect kth power and if gcd(ai,aj)=1 for i≠j , then each ai is a perfect kth power.