Prove that 5 is not a rational number You may assume that if

Prove that 5 is not a rational number. (You may assume that if 5 divides x 2 then 5 divides x for any integer x.)

Solution

If 5 is rational, then it can be expressed by some number a/b (in lowest terms). This would mean:
(a/b)² = 5. Squaring,
a² / b² = 5. Multiplying by b²,
a² = 5b².

If a and b are in lowest terms (as supposed), their squares would each have an even number of prime factors. 5b² has one more prime factor than b², meaning it would have an odd number of prime factors.

Every composite has a unique prime factorization and can\'t have both an even and odd number of prime factors. This contradiction forces the supposition wrong, so 5 cannot be rational. It is, therefore, irrational.