Discussion
Let x, y and z be integers such that 7x + 2y - 5z is divisible by 11. What is the remainder when 3x - 7y + 12z is divided by 11?
Answer: A
The remainder is 0. Multiply the given expression by 2: \[2(7x + 2y - 5z) = 14x + 4y - 10z\] Reducing the coefficients modulo 11 gives 14 = 3 + 11, 4 stays 4 and -10 = 1 - 11, so \[14x + 4y - 10z \equiv 3x + 4y + z \pmod{11}\] Now -7y is congruent to 4y and 12z to z modulo 11, so \(3x - 7y + 12z \equiv 3x + 4y + z\). Therefore 3x - 7y + 12z is congruent to twice a multiple of 11, and the remainder is 0. The options 1, 5 and 7 would need a different relationship between the two expressions.
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