If positive integers a and b are both odd,

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Vincen: Legendary Member; Posts: 2898; Joined: Thu Sep 07, 2017 2:49 pm; Thanked: 6 times; Followed by:5 members

If positive integers a and b are both odd,

by Vincen » Mon Sep 25, 2017 6:57 pm

If positive integers a and b are both odd, which of the following must not be odd?

A) ab
B) a(ab)
C) b - a - 1
D) (a + b)/2
E) a - b

The OA is E.

If we set a=3 and b=5 the (a+b)/2 is 4 and it is not odd. Why option D is wrong?

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pannalal: Junior | Next Rank: 30 Posts; Posts: 22; Joined: Wed Sep 20, 2017 5:42 pm; Thanked: 2 times

by pannalal » Tue Sep 26, 2017 4:58 am

Vincen wrote:If positive integers a and b are both odd, which of the following must not be odd?

A) ab
B) a(ab)
C) b - a - 1
D) (a + b)/2
E) a - b

The OA is E.

If we set a=3 and b=5 the (a+b)/2 is 4 and it is not odd. Why option D is wrong?

Given a and b are both odd. Just put a = 1 and b = 1, then check all the options:
(A) ab = 1*1 = 1 Odd
(B) a*(ab) = 1*(1*1) = 1 Odd
(C) b - a - 1 = 1 - 1 - 1 = -1 Odd
(D) (a+b)/2 = (1+1)/2 = 1 Odd
(E) a - b = 1 - 1 = 0 Even

So, the answer is E

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ErikaPrepScholar: Legendary Member; Posts: 503; Joined: Thu Jul 20, 2017 9:03 am; Thanked: 86 times; Followed by:15 members; GMAT Score:770

by ErikaPrepScholar » Tue Sep 26, 2017 8:33 am

The most important thing about this question is that it asks for the answer that must not be odd. This means that an answer will only be right if it is always odd, no matter what odd numbers we plug in for a and b. An answer will be wrong if it is 1) always even OR 2) sometimes odd, sometimes even.

So you're totally right that plugging 3 and 5 into D gives an even number. HOWEVER, plugging 1 and 5 (both odd numbers) into D gives

(1+5)/2
6/2
3

Which is odd. So D can be odd, which means that it is not correct to say that it must not be odd.

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by [email protected] » Tue Sep 26, 2017 11:17 am

Hi Vincen,

This question can be approached with Number Property rules. We're told that A and B are both ODD, POSITIVE INTEGERS. We're asked which of the following MUST NOT be odd.

Answer A: (A)(B) = (odd)(odd) = odd
Answer B: (A)(A)(B) = (odd)(odd)(odd) = odd
Answer C: B - A - 1 = (odd) - (odd) - (odd) = odd
Answer D: (A+ B)/2 = (odd + odd)/2 = (even)/2 = COULD be even (re: 4/2 = 2) OR odd (re: 6/2 = 3)
Answer E: A - B = (odd) - (odd) = EVEN

There's only one answer that can NEVER be odd.

Final Answer: E

GMAT assassins aren't born, they're made,
Rich

Contact Rich at [email protected]

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Matt@VeritasPrep: GMAT Instructor; Posts: 2630; Joined: Wed Sep 12, 2012 3:32 pm; Location: East Bay all the way; Thanked: 625 times; Followed by:119 members; GMAT Score:780

by Matt@VeritasPrep » Tue Sep 26, 2017 4:50 pm

You asked about D specifically, so I'll only address that.

It actually comes down to the remainder of our odd integers when divided by 4. If they have different remainders (one has remainder 1, one has remainder 3), then the average of the two will be even. If they have the same remainder by 4 (both remainder 1 or both remainder 3), then the average will be odd.

A few examples:

5 + 7 => 12, average = 6
5 + 9 => 14, average = 7
7 + 11 => 11, average = 9

As a good follow up exercise, see if you can explain why this must be. (Hint: a multiple of 4 can be written as 4*some integer.)

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