Is it possible to prove directly that even perfect squares have even square roots? Or, symbolically: forall n in Z, n^2 is even =>n is even

Diana Suarez 2022-09-27 Answered
Is it possible to prove directly that even perfect squares have even square roots? Or, symbolically:
n Z ,     n 2  is even  n  is even  n Z ,     n 2  is even  n  is even 
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Answers (2)

GrEettarim3
Answered 2022-09-28 Author has 6 answers
Since 2 is a prime, and since p | a b implies either p | a or p | b, that tells you that 2 | n 2 implies that 2 | n.
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Ryland Houston
Answered 2022-09-29 Author has 4 answers
Let n = m 2 . Let m = p i n i be the unique prime factorization of m. Then n 2 = p i 2 n i is the prime factorization of n 2 . As n 2 is even one of those primes is 2. Those are the primes that factor m so m is even.
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