Question

If $n = 1 + x$ where $x$ is the product of four consecutive positive integers, then which of the following is/are true? A. $n$ is odd B. $n$ is prime C. $n$ is a perfect square

Accepted Answer

A. Let $x = k(k+1)(k+2)(k+3)$. Note that $x = (k^2+3k)(k^2+3k+2) = m(m+2) = m^2 + 2m$. So $n = 1 + x = m^2 + 2m + 1 = (m+1)^2 = (k^2+3k+1)^2$. Thus $n$ is always a perfect square. Also, among four consecutive integers, two are even and two are odd, so $x$ is even, making $n = 1+x$ odd. Since $n$ is a square of an integer $>1$, $n$ cannot be prime. Hence A and C are true.

Properties of Integers: Product of 4 Consecutive Integers

Explanation

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