Two positive integers leave the SAME remainder when each is divided by d. Then d must always divide:

Write a = dq₁ + r and b = dq₂ + r with the same r. Subtracting, a − b = d(q₁ − q₂), so d divides a − b. It need not divide a or b themselves (17 and 32 both leave remainder 2 on division by 5, yet 5 divides neither), nor their sum or product in general. This is exactly the trick used in every “greatest number leaving remainder r” problem.