Problem Solving

If a and b are positive integers such that a – 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. (2+b) /2

a-b is an even integer

This is possible only when a and b both are either even or odd
i.e. a = 10 and b = 4
or a = 9 and b = 5

a/b is an even integer

Since we can have both odd or both even a/b is even only when a and b, both are even...

Now from the ans choices...

(a+b) / 2 will provide the odd ans..

Ans C.

The best strategy is to pick numbers

Since its a must question try plugging in more than 3 values for a and b

I think the answer is D because for all other values there is a yes and a no as an answer

The answer is not C because lets assume a=8 and b=4 then (a+b)/2=12/2=6 which is an even number.

Is the answer D??

yeah answer is D thanks

moneyman wrote:The best strategy is to pick numbers

Since its a must question try plugging in more than 3 values for a and b

I think the answer is D because for all other values there is a yes and a no as an answer

The answer is not C because lets assume a=8 and b=4 then (a+b)/2=12/2=6 which is an even number.

Is the answer D??

can't the answer also be E since 4+2=6 and 6/2=3?

hummm???

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

a-b is even so both a& b are either odd or even

a/b is even, possible only if both are even(considering the above case also into account)

so we can say that both a & b are even numbers

also a/b =c (c is even) i.e a = bc

now we have

A a/2 = bc/2 is even ( both b & c have 2 as a factor)
B. b/2 = odd or even
C. (a+b) /2 = (bc + b)/2 = b(c+1)/2 (may be even or odd)
D. (a+2) /2 = a/2 + 1 = even nos + 1 = odd
E. (2+b) /2 = 1 + b/2 (b/2 can be both even as well as odd)

I dont see how D can be the answer:

"a" and "b" must be both even integers

D. (a+2)/2 must be odd

Say that "a" is 10 (an even integer)....10+2=12......12/2=6 (and is an even integer)

Its D.. as it is given that a/b is even means it cannot be 10/2 as 5 is odd... so a can be 8 or 12.
Then (a+2)/2 is always odd.

Got it!!...guess that with the gmat nothing is as simple as ""a" and "b" are both even"...i will try to keep that in mind

Thanks rchadha7!

Got it!!...guess that with the gmat nothing is as simple as ""a" and "b" are both even"...i will try to keep that in mind

Thanks rchadha7!

a - b = 2m where m GE 0. --(1)
a = 2qb where q GE 0 -- (2)

From (2) a is even. Hence b is also even from (1).

now (a + 2)/2 = (2qb + 2)/2 = qb + 1

b is even implies qb is even implies qb + 1 must be ODD.

Hence (a+2)/2 is odd.

Calista.