thevenus wrote:

If a, b, and c are positive integers, what is the remainder when a – b is divided by 6?

(1) a = c^3

(2) b = (c – 2)^3

Manhattan GMAT weekly challenge (2nd week, Oct,2012)

Obviously, neither (1), nor (2) alone is sufficient.

Integers, when divided by 6 leave remainders 0, 1, 2, 3, 4, or 5.

When raised to the third power, the remainders stay the same 0, 1, 2, 3, 4, or 5:

\(6^3\)- remainder 0

\(1^3\) - remainder 1

\(2^3=8=6+2\) - remainder 2

\(3^3=27=24+3\) - remainder 3

\(4^3=64=60+4\) - remainder 4

\(5^3=125=120+5\) - remainder 5

Therefore, when subtracting the cubes of two integers two units apart, the difference will leave a remainder of 2 when divided by 6:

\(2-0=3-1=4-2=5-2\) and \(0-4=1-5=-4=-6+2.\)

Remainders repeat themselves cyclically \(0,1,2,3,4,5,0,1,2,3,4,5...\)

Answer C.

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PhD in Applied Mathematics

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