airan wrote:x,a and b are positive integers. When x is divided by a, the remainder is b. When x is divided by b, the remainder is a-2. Which of the following must be true.

A. A is even.
B. x+b is divisible by a.
C. x-1 is divisible by a.
D. b=a-1
E. a+2=b+1

From the statements, what jumps out at me first is that a and b need to be very close in value. Recall that when you divide by, say, 7, the only possible remainders are 0, 1, 2, 3, 4, 5 and 6. That is, the remainder must be less than 7 when you divide by 7, and similarly, the remainder must be less than z when you divide by z. Onto the question:

When x is divided by a, the remainder is b.

So b < a

When x is divided by b, the remainder is a-2.

So a-2 < b.

Thus a-2 < b < a, and since b is an integer, b = a-1.