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

Statement 1 does not state anything about b, and Statement 2 does not state anything about a, so the answer can be either C or E.

b - a = (c + 1)^3 - c^3 = (c^3 + 3c^2 + 3c + 1) - c^3 = 3c^3 + 3c + 1 = 3(c^2 + c) + 1

=3(c^2 + c) + 1] / 3 = (c^2 + c) + 1/3

The remainder = 1. Sufficient.

The correct answer: C

Hope this helps!

-Jay
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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.