a, b, c, d are positive integers. When a is divided by b, the remainde

Recall that the remainder must be less than the divisor; thus, b must be greater than 7 since “when a is divided by b, the remainder is 7.” Similarly, d must be greater than 5 since “when c is divided by d, the remainder is 5.” Since b and d are positive integers, b ≥ 8 and d ≥ 6.

Next we can factor each answer choice. If the number can be factored into two numbers such that one of them is at least 8 and the other is at least 6, then it can be a value of b x d. However, if the number can’t be factored in the way described above, then it CANNOT be a value of b x d and that is the number we are looking for.

A. 48 = 8 x 6 (Yes)
B. 56 = 8 x 7 (Yes)
C. 63 = 9 x 7 (Yes)
D. 69 = 1 x 69 = 3 x 23 (No)
E. 140 = 20 x 7 (Yes)

Since 69 can only be factored as 1 x 69 or 2 x 23, we don’t have one factor ≥ 8 and the other ≥ 6, so 69 can’t be a value of b x d.

Answer: D