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Manager  Joined: 17 Aug 2009
Posts: 144
If a and b are positive integers such that a – b and a/b are  [#permalink]

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Difficulty:   95% (hard)

Question Stats: 46% (02:19) correct 54% (02:26) wrong based on 870 sessions

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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. (b+2)/2
Math Expert V
Joined: 02 Sep 2009
Posts: 58989
If a and b are positive integers such that a – b and a/b are  [#permalink]

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28
23
zaarathelab wrote:
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. (b+2)/2

$$a-b$$ even --> either both even or both odd

$$\frac{a}{b}$$ even --> either both even or $$a$$ is even and $$b$$ is odd.

As both statements are true --> $$a$$ and $$b$$ must be even.

As $$\frac{a}{b}$$ is an even integer --> $$a$$ must be multiple of 4.

Options A is always even.

Options B can be even or odd.

Options C can be even or odd.

Options D: $$\frac{a+2}{2}=\frac{a}{2}+1$$, as $$a$$ is multiple of $$4$$, $$\frac{a}{2}$$ is even integer --> even+1=odd. Hence option D is always odd.

Options E can be even, odd.

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Math Expert V
Joined: 02 Sep 2009
Posts: 58989
Re: Even and Odd : GMATPrep  [#permalink]

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sagarsabnis wrote:
I didnt get why a has to be multiple of 4

Now if u take a as 4 and b as 2 then a-b is even a/b is even and a+b/2 is odd which is option C.

Please explain where i am going wrong?

First question: Why $$a$$ has to be multiple of 4?

As we concluded $$a$$ and $$b$$ have to be even integers to meet both conditions in the stem: $$a=2m$$ and $$b=2n$$.

Then we have $$\frac{a}{b}$$ is an even integer: $$\frac{a}{b}=2k$$ --> $$a=2bk=4kn$$ --> $$a$$ is a multiple of $$4$$.

Second question: Option C also COULD give an even result, so why not C?

The question asks: "which of the following MUST be an odd integer?"

If we take $$a=16$$ and $$b=4$$, then $$\frac{a+b}{2}=10=even$$. So option C may or may not be odd and we are asked to determine which option is ALWAYS odd, hence C is out.

In fact:
Options A is always even.
Options B can be even or odd.
Options C can be even or odd.
Options D is always odd.
Options E can be even or odd.

Only option which is always odd is D.

Hope it helps.
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Re: If a and b are positive integers  [#permalink]

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1
Given:
a > 0, b > 0 and a, b are Integers

a - b and a/b are both even intergers
==> $$a - b = 2m_1$$
$$\frac{a}{b} = 2m_2$$ ==> $$a = (b)(2m_2)$$
==> a & b are both even, otherwise the above conditions won't be satisfied.

Now lets go with options one by one ...
A) a/2
==> can be even or odd ( a = 6 or a = 8)

B) b/2
==> Can be either even or odd ( b = 6 or b = 8)

C) (a+b) /2
==> Since, $$a = (b)(2m_2)$$, $$\frac{a + b}{2}$$ =$$\frac {(b)(2m_2) + b}{2}$$
==> $$\frac{a + b}{2}$$ =$$\frac {(b)(2m_2 + 1)}{2}$$
==> Can be even or odd

D) (a+2)/2
==> $$\frac{a + b}{2}$$ =$$\frac{2bm_2 + 2}{2}$$
==> $$\frac{a + b}{2}$$ =$$bm_2 + 1$$
==> Since, b is even, $$bm_2$$ is even and $$bm_2 + 1$$ is odd always

E) (b+2)/2
==> $$\frac{b+2}{2} = 1 + \frac{b}{2}$$
==> Can be even or odd
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Re: If a and b are positive integers such that a – b and a/b are  [#permalink]

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zaarathelab wrote:
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. (b+2)/2

Given that (a-b) = even

Thus, $$b(\frac{a}{b}-1)$$ = even --> as $$\frac{a}{b}$$ is even, $$(\frac{a}{b}-1)$$ will be odd. Thus, b*odd = even--> b must be even

Thus, as a = b*2p --> a = 2l*2p = 4lp , where l,p are non-zero positive integers.

Thus,$$\frac{a}{2}$$ + 1 = 2lp+1, which will always be an odd number.

D.
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Manager  Joined: 17 Aug 2009
Posts: 144
Re: Even and Odd : GMATPrep  [#permalink]

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Quote:
As \frac{a}{b} is an even integer --> a must be multiple of 4.

Bunuel, you are an absolute whiz

Yep the catch is that a has to be a multiple of 4 for a/b to be even

OA is D
Manager  Joined: 22 Jul 2009
Posts: 112
Location: Manchester UK
Re: Even and Odd : GMATPrep  [#permalink]

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I didnt get why a has to be multiple of 4

Now if u take a as 4 and b as 2 then a-b is even a/b is even and a+b/2 is odd which is option C.

Please explain where i am going wrong?
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Re: If a and b are positive integers such that a – b and a/b are  [#permalink]

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zaarathelab wrote:
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. (b+2)/2

The video solution is explained here.

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Re: If a and b are positive integers such that a-b and a/b are  [#permalink]

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_________________ Re: If a and b are positive integers such that a-b and a/b are   [#permalink] 28 Aug 2019, 23:30
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