If \(x, a,\) and \(b\) are positive integers such that when \(x\) is divided by \(a,\) the remainder is \(b\) and when

If \(x, a,\) and \(b\) are positive integers such that when \(x\) is divided by \(a,\) the remainder is \(b\) and when \(x\) is divided by \(b,\) the remainder is \(a-2,\) then 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. \(a+2=b+1\)

Answer: D

Source: GMAT Club Tests