For the positive integers q, r, s, and t,

Vincen wrote:For the positive integers q, r, s, and t, the remainder when q is divided by r is 7 and the remainder when s is divided by t is 3. All of the following are possible values for the product rt EXCEPT:

A. 32
B. 38
C. 44
D. 52
E. 63

OA is B.

To solve this problem, we need a clear insight on division between integers. Recall that in any division, the divisor must be greater than the remainder. Thus, we know that r > 7 and t > 3. In the other words, r ≥ 8 and t ≥ 4. Thus, rt is at least 8(4) = 32. Looking at the answer choices, all of them are at least 32. Therefore, we need further analysis. One thing we can do is to factor each answer choice:

32 = 1(32) = 2(16) = 4(8)

38 = 1(38) = 2(19)

44 = 1(44) = 2(22) = 4(11)

52 = 1(52) = 4(13)

63 = 1(63) = 3(21) = 7(9)

We see that all the answer choices except 38 can be expressed as a product of two factors where one factor is greater than 7 and the other is greater than 3. For example, rt could be 44 since r could be 11 and t could be 4. The only answer choice that doesn’t have this property is 38. Thus, 38 is the correct answer.

Answer: B