If x is a rational number such that \(x^2\) is an integer, which of the following must be true?
x is always an integer
Let \(x = \dfrac{p}{q}\) be rational in lowest terms, so \(\gcd(p, q) = 1\).
Then \(x^2 = \dfrac{p^2}{q^2}\), and since \(\gcd(p^2, q^2) = 1\) as well, this fraction is already in lowest terms.
For \(x^2\) to be an integer, the denominator \(q^2\) must equal 1, which forces \(q = 1\).
Therefore x itself must always be an integer.
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