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Math Expert V
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If a and b are integers, is a + b + 3 an odd integer?  [#permalink]

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Question Stats: 76% (01:18) correct 24% (01:21) wrong based on 1260 sessions

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If a and b are integers, is a + b + 3 an odd integer?

(1) ab is an odd integer.
(2) a − b is an even integer.

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If a and b are integers, is a + b + 3 an odd integer?  [#permalink]

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1
Bunuel wrote:
If a and b are integers, is a + b + 3 an odd integer?

(1) ab is an odd integer.
(2) a − b is an even integer.

(1) $$ab$$ is an odd integer.

$$Odd * Odd = Odd$$

Therefore $$a$$ and $$b$$ should be Odd integers.

($$Odd + Odd = Even$$ ) and ($$Even + Odd = Odd$$)

$$a + b = Even$$

$$a + b + 3 = Even + 3$$ = Odd integer.

Hence I is Sufficient.

(2) $$a − b$$ is an even integer.

$$Odd - Odd = Even$$

$$Even - Even = Even.$$

Therefore $$a$$ and $$b$$ could both either be Even or both be Odd.

If $$a$$ and $$b$$ are Odd. ($$Odd + Odd = Even$$) and ($$Even + Odd = Odd$$)

$$a + b + 3 = Odd + Odd + 3 = Even + 3 = Odd$$ --------- True.

If $$a$$ and $$b$$ are Even. ($$Even + Even = Even$$) and ($$Even + Odd = Odd$$)

$$a + b + 3 = Even + Even + 3 = Even + 3 = Odd$$ ------- True.

Hence II is Sufficient. Answer (D)...
##### General Discussion
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If a and b are integers, is a + b + 3 an odd integer?  [#permalink]

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1
Given : a and b are integers
We need to find out if a + b + 3 is an odd integer

(1) ab is an odd integer.
This is possible only when both a and b are odd.
Hence sum of a and b will be even, making the sum(a + b + 3) an odd integer(Sufficient)
(2) a − b is an even integer.
This is possible when both a and b is even or odd.
In either case, the sum of a and b is even.
Thus, a+b+3 is odd(Sufficient) (Option D)
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Re: If a and b are integers, is a + b + 3 an odd integer?  [#permalink]

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If a and b are integers, is $$a + b + 3$$ an odd integer?

In order to find $$a + b + 3$$ is ODD, we need to find.

ODD + ODD + 3 = ODD

EVEN + EVEN + 3 = ODD

Which means, if we are able to confirm if a & b both are ODD or if a & b both are EVEN we should be able to answer the question.

(1) ab is an odd integer

ab = ODD

This means that a & b both are ODD

As per above this information should be sufficient to prove $$a + b + 3$$ is ODD as ODD + ODD + 3 = ODD

Hence, Eq. (1) =====> is SUFFICIENT

(2) a − b is an even integer

a - b is EVEN

This is only possible if

a & b are both ODD

a & b are both EVEN

As this is sufficient to answer the question $$a + b + 3$$ is ODD

Hence, Eq. (2) =====> is SUFFICIENT

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Re: If a and b are integers, is a + b + 3 an odd integer?  [#permalink]

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2
Given that a and b are both integers, so each of them must be either even or odd. a+b+3 will be odd when (a+b) is even.

Statement 1. product of a & b is odd, which is only possible when both a & b are odd. And if both a & b are odd, then (a+b) is even. So we know. Sufficient.

Statement 2. a-b is even, this will happen either when
both a & b are even, in which case (a+b) will also be even
or when both a & b are odd, in which case too (a+b) will be even.

a+b is even in any case, so we know. Sufficient.

Each statement alone is sufficient. Hence D answer
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Re: If a and b are integers, is a + b + 3 an odd integer?  [#permalink]

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1
Bunuel wrote:
If a and b are integers, is a + b + 3 an odd integer?

(1) ab is an odd integer.
(2) a − b is an even integer.

for a+b+3 to be odd a+b should be even

1) if ab is an odd integer then a and b are individually odd, so a+b= odd+odd=Even : sufficient .
2)a- b is an even integer, so a and b are both even or both odd. In either cases the sum a+b is even : sufficient.
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Re: If a and b are integers, is a + b + 3 an odd integer?  [#permalink]

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Bunuel wrote:
If a and b are integers, is a + b + 3 an odd integer?

(1) ab is an odd integer.
(2) a − b is an even integer.

We need to determine whether a + b + 3 is odd, or in other words, whether a + b is even.

Statement One Alone:

ab is an odd integer.

Because ab is odd, we know that a and b must both be odd, and since odd + odd = even, a + b is even. Statement one alone is sufficient to answer the question.

Statement Two Alone:

a - b is an even integer.

Since a - b is even, a + b is also even. Statement two alone is sufficient.

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Re: If a and b are integers, is a + b + 3 an odd integer?  [#permalink]

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While I am clear on the even and odd concept ,for this question given that they are integers- do we ignore about them being positive or negative integers?
Math Expert V
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Posts: 61526
Re: If a and b are integers, is a + b + 3 an odd integer?  [#permalink]

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deeptijp wrote:
While I am clear on the even and odd concept ,for this question given that they are integers- do we ignore about them being positive or negative integers?

a and b could be negative even/odd integers but this does not change the answer. Is it what you were asking?
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Re: If a and b are integers, is a + b + 3 an odd integer?  [#permalink]

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Bunuel wrote:
deeptijp wrote:
While I am clear on the even and odd concept ,for this question given that they are integers- do we ignore about them being positive or negative integers?

a and b could be negative even/odd integers but this does not change the answer. Is it what you were asking?

yes,as I was a bit doubtful when solving the question that when do we consider the signs as no one posted solution considering signs.
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Re: If a and b are integers, is a + b + 3 an odd integer?  [#permalink]

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amanvermagmat niks18

Quote:
Given that a and b are both integers, so each of them must be either even or odd. a+b+3 will be odd when (a+b) is even.

Statement 1. product of a & b is odd, which is only possible when both a & b are odd. And if both a & b are odd, then (a+b) is even. So we know. Sufficient.

Statement 2. a-b is even, this will happen either when
both a & b are even, in which case (a+b) will also be even
or when both a & b are odd, in which case too (a+b) will be even.

a+b is even in any case, so we know. Sufficient.

Each statement alone is sufficient. Hence D answer

I landed with same ans using identical approach of dissecting Q stem before moving to ans choices.
I just took few additional secs for checking for St 2 by taking random nos. Did you carry above step mentally?
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Re: If a and b are integers, is a + b + 3 an odd integer?  [#permalink]

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amanvermagmat niks18

Quote:
Given that a and b are both integers, so each of them must be either even or odd. a+b+3 will be odd when (a+b) is even.

Statement 1. product of a & b is odd, which is only possible when both a & b are odd. And if both a & b are odd, then (a+b) is even. So we know. Sufficient.

Statement 2. a-b is even, this will happen either when
both a & b are even, in which case (a+b) will also be even
or when both a & b are odd, in which case too (a+b) will be even.

a+b is even in any case, so we know. Sufficient.

Each statement alone is sufficient. Hence D answer

I landed with same ans using identical approach of dissecting Q stem before moving to ans choices.
I just took few additional secs for checking for St 2 by taking random nos. Did you carry above step mentally?

On adding or subtracting two integers, the nature of resulting value will not change I.e if on subtracting two integers the resulting value is Even then on adding the same numbers the resulting value will be Even. If you thought about this then the question is very simple and you might not need to test values.
So if a=5 and b=3, then a-b=2 and a+b=8 both are Even

Similarly if both are Even then result will obviously be Even

Posted from my mobile device
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If a and b are integers, is a + b + 3 an odd integer?  [#permalink]

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amanvermagmat niks18

Quote:
Given that a and b are both integers, so each of them must be either even or odd. a+b+3 will be odd when (a+b) is even.

Statement 1. product of a & b is odd, which is only possible when both a & b are odd. And if both a & b are odd, then (a+b) is even. So we know. Sufficient.

Statement 2. a-b is even, this will happen either when
both a & b are even, in which case (a+b) will also be even
or when both a & b are odd, in which case too (a+b) will be even.

a+b is even in any case, so we know. Sufficient.

Each statement alone is sufficient. Hence D answer

I landed with same ans using identical approach of dissecting Q stem before moving to ans choices.
I just took few additional secs for checking for St 2 by taking random nos. Did you carry above step mentally?

Hi

Yes, now I dont need to do any calculation mentally regarding even/odd in these kind of questions - because the basic rules regarding addition/subtraction/multiplication of even/odd are drilled in my head.
I suggest that you too look at these rules again (from number theory) here (in case you think you need to improve on this):

https://gmatclub.com/forum/math-number- ... 88376.html

And then practice with as many values as you can, till it becomes easy and natural to you. Good luck!
Director  D
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If a and b are integers, is a + b + 3 an odd integer?  [#permalink]

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Hi,

Statement 1 indicates both are odd whereas statement 2 indicates both are even.

Is it possible that a and b both are of different type of no. (odd and even) together to satisfy the single condition, I.e, a+b+3=odd.

Kindly throw some lights.

Regards,
Raxit T.

Bunuel wrote:
deeptijp wrote:
While I am clear on the even and odd concept ,for this question given that they are integers- do we ignore about them being positive or negative integers?

a and b could be negative even/odd integers but this does not change the answer. Is it what you were asking?
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Re: If a and b are integers, is a + b + 3 an odd integer?  [#permalink]

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Raxit85 wrote:
Hi,

Statement 1 indicates both are odd whereas statement 2 indicates both are even.

Is it possible that a and b both are of different type of no. (odd and even) together to satisfy the single condition, I.e, a+b+3=odd.

Kindly throw some lights.

In statement two both, a and b can also be odd since odd - odd = even
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Re: If a and b are integers, is a + b + 3 an odd integer?  [#permalink]

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We need to determine whether a + b + 3 is odd or, in other words, whether a + b is even.

Statement One Alone:

ab is an odd integer.

Thus, a and b are both odd, and since odd + odd = even, a + b is even. Statement one alone is sufficient to answer the question.

Statement Two Alone:

a - b is an even integer.

Since a - b is even, a + b is also even. Statement two alone is sufficient.
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Re: If a and b are integers, is a + b + 3 an odd integer?  [#permalink]

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ScottTargetTestPrep wrote:
Bunuel wrote:
If a and b are integers, is a + b + 3 an odd integer?

(1) ab is an odd integer.
(2) a − b is an even integer.

We need to determine whether a + b + 3 is odd, or in other words, whether a + b is even.

Statement One Alone:

ab is an odd integer.

Because ab is odd, we know that a and b must both be odd, and since odd + odd = even, a + b is even. Statement one alone is sufficient to answer the question.

Statement Two Alone:

a - b is an even integer.

Since a - b is even, a + b is also even. Statement two alone is sufficient.

In statement 2 , if even -even is even and odd- odd is even , hence if a and b are even and if if a & B are odd , final result a+b+3 will change .The how it is sufficient?
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Re: If a and b are integers, is a + b + 3 an odd integer?  [#permalink]

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rinkipanwar123 wrote:
ScottTargetTestPrep wrote:
Bunuel wrote:
If a and b are integers, is a + b + 3 an odd integer?

(1) ab is an odd integer.
(2) a − b is an even integer.

We need to determine whether a + b + 3 is odd, or in other words, whether a + b is even.

Statement One Alone:

ab is an odd integer.

Because ab is odd, we know that a and b must both be odd, and since odd + odd = even, a + b is even. Statement one alone is sufficient to answer the question.

Statement Two Alone:

a - b is an even integer.

Since a - b is even, a + b is also even. Statement two alone is sufficient.

In statement 2 , if even -even is even and odd- odd is even , hence if a and b are even and if if a & B are odd , final result a+b+3 will change .The how it is sufficient?

In statement 2 , if even -even is even and odd- odd is even , hence if a and b are even and if if a & B are odd , final result a+b+3 will change .The how it is sufficient?

If a and b are even, then a + b is even. If a and b are odd, then again a + b is even. The question is asking for us to determine whether a + b + 3 is odd; since a + b is even, a + b + 3 will be odd in either case. That’s why the statement “a - b is even” is sufficient to determine whether a + b + 3 is odd.
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If you find one of my posts helpful, please take a moment to click on the "Kudos" button. Re: If a and b are integers, is a + b + 3 an odd integer?   [#permalink] 05 Feb 2020, 11:19
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