Is 2(2k-1) is a perfect square for positive integer k?

Question

For positive integer $k$, let $M = 2(2k-1)$, which of the following must be true?

(a) $M$ is not a perfect square for any $k$.

(b) There are infinitely many $k$ such that $M$ is a perfect square.

(c) There is a unique $k$ such that $M$ is a perfect square.

(d) There are finitely many, but more than 1, values of $k$ such that $M$ is a perfect square.

This question was on my Math Challenge II Number Theory packet. I'd also like a proof of how you got your answer.

Answer 1

What are some properties of perfect squares? What do you know about the factors of $M$? The truth value of the statements should follow quite easily.

Answer:

$M$ is always a product of an even and odd number, so it only has one factor of 2, and thus can't be a perfect square, since if it were, it must have a factor of at least $2^2 = 4$.