When 900 is divided by positive integer d, the remainder is

[BTGmoderatorDC]

When 900 is divided by positive integer d, the remainder is r. For some integer N > 5000, when N is divided by positive integer D, the remainder is R. Is R > d?

(1) r = 1

(2) D = 23

OA C

Source: Magoosh

[Ian Stewart]

Each Statement is clearly insufficient alone, because using only one statement, we know almost nothing about one of the two unknowns in the inequality in the question.

Using Statement 1, we know when 900 is divided by d, the remainder is 1. That means that 900 is exactly 1 larger than some multiple of d, or in other words, 900-1 = 899 is divisible by d. Now 899 = 900 - 1 = 30^2 - 1^2 = (30+1)(30-1) = 31*29, and since these are both prime, d can only be equal to 29, 31 or 899 (d cannot be 1, the only other factor of 899, since if d=1, the remainder would be 0 when 900 is divided by d). Using Statement 2, we know that R is a remainder we can get when dividing by 23, so R must be an integer between 0 and 22 inclusive. Since R < 22 and d > 29, R < d must be true, and the answer to the question must be "no", so the two statements are sufficient together and the answer is C.