If p and q are positive integers, and the remainder obtained when p is

The remainder obtained when p is divided by q is the same as the remainder obtained when q is divided by p
This information is indirectly telling us that p = q
To explain why, let's see what happens if p does NOT equal q
If that's the case, then one value must be greater than the other value.
Let's see what happens IF it were the case that p < q.

What is the remainder when p is divided by q?
Since p < q, then p divided by q equals 0 with remainder p

IMPORTANT RULE: When positive integer N is divided by positive integer D, the remainder R is such that 0 ≤ R < D
For example, if we divide some positive integer by 7, the remainder will be 6, 5, 4, 3, 2, 1, or 0

What is the remainder when q is divided by p?
Based on the above rule, we know that the remainder must be a number such that 0 ≤ remainder < p

Hmmmmm. In our first calculation (p ÷ q), we found that the remainder = p
In our second calculation (q ÷ p), we found that 0 ≤ remainder < p
Since it's IMPOSSIBLE for the remainder to both EQUAL p and BE LESS THAN p, we can conclude that it's impossible for p to be less than q.

Using similar logic, we can see that it's also impossible for q to be less than p.

So, it MUST be the case that p = q
So, pq = p² = the square of some integer

Check the answer choices . . . only D is the square of an integer.

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