Suppose n, a are integers. If n divides a, then n divides a^2.
To prove this statement, let's start by assuming that n divides a. This means that there exists an integer k such that a = nk.
Taking the square of both sides, we have (a)^2 = (nk)^2. Expanding the right side of the equation, we get a^2 = n^2 * k^2.
Since both n^2 and k^2 are integers, we can see that a^2 is divisible by n^2. Therefore, if n divides a, then n also divides a^2.
To summarize the proof:
1. Assume that n divides a.
2. Express a as a = nk for some integer k.
3. Square both sides: (a)^2 = (nk)^2.
4. Expand: a^2 = n^2 * k^2.
5. Since n^2 and k^2 are integers, n^2 divides a^2.
6. Hence, if n divides a, then n also divides a^2.