M8 wrote:

If a and b are positive integers such that a â€“ b and a/b are both even integers, which of the following must be an odd integer?

A. a/2

B. b/2

C. (a+b)/2

D. (a+2)/2

E. (b+2)/2

Please explain your solution.

doesn't work when a is odd because a/b will be always odd

so a must be even, => b is even

pick numbers

(A) a/2

a could be 4

b could be 2

a-b and a/b are even hold

a/2 = 2,

so this is not the answer

(B) b/2

a could be 8

b could be 4

a-b and a/b are even hold

b/2 = 4

so this is not the answer

(C)

a could be 8

b could be 4

a-b and a/b are even hold

(a+b)/2 = 6

so this is not the answer

(E)

a could be 4

b could be 2

a-b and a/b are even hold

(b+2)/2 = 2

so this is not the answer

so we are left with:

(D) (a+2)/2

by elimination this is the answer a proper proof might be to time consuming, so just stick to elimination.

however for a-b and a/b to be even integers at the same time a-b must be a multiple of 4, if you add 2 to a then it becomes the middle point between two even, so is an odd