BTGmoderatorDC wrote:If a, b, and c are positive integers, what is the remainder after b - a is divided by 3?
(1) a = c^3
(2) b = (c + 1)^3
OA C
Source: Manhattan Prep
Another option is to test values. As Jay said, we need information about both b and a, so neither statement will be sufficient on its own. When we combine the statements, we can test values:
Case 1:
c = 1
a = 1^3 = 1
b = 2^3 = 8
b - a = 8 - 1 = 7
the remainder when 7 is divided by 3 is 1.
Case 2:
c = 2
a = 2^3 = 8
b = 3^3 = 27
b - a = 27 - 8 = 19
the remainder when 19 is divided by 3 is 1.
If you're pressed for time, just guess C and move on. If you have time, try 1 more case just to be sure:
Case 3:
c = 3
a = 3^3 = 27
b = 4^3 = 64
b - a = 64 - 27 = 37
the remainder when 37 is divided by 3 is 1.
If we get the same result in 3 cases (where there are restrictions on the kinds of cases we can test; negatives, zero, and fractions weren't options here), we can trust that there is a consistent pattern, and the answer must be
C.