Lesson: Even and Odd Integers

Comment on Even and Odd Integers

0 is neither even nor odd.
gmat-admin's picture

Zero is even.
An integer is even if that integer is divisible by 2. An integer, n, is divisible by 2 if n = 2k, where k is an integer.

So, for example, 10 is even because 10 = (2)(5), and 5 is an integer.
Likewise, -22 is even because -22 = (2)(-11), and -11 is an integer.
Similarly, 0 is even because 0 = (2)(0), and 0 is an integer.

zero is neither positive nor negative
gmat-admin's picture

That's true

appears this lesson (and the reinforcement ques associated with the video) are arguably one of the most important ones in your module in preparation for the test. do you have a top 10 must know inside and out videos (from GMATPrepNow), based on your knowledge of where the 37 quant questions might come from
gmat-admin's picture

I don't have such a list. However, the number of practice questions related to each topic is a good indication of the importance of that topic.

Hi Brent,

Question: is it true that fractions and or decimals are not considered Odd or Even numbers?
One of the post answered by some user mentioned that a number like 1.3 cannot be Odd or Even because the addition of a zero at the end of the number (which does not change the value) makes it even divisible by "2" whereas 1.3 would be Odd. Maybe I am interpreting the information the wrong way or the user was wrong or failed to explain adequately. Let me know

Thank you
gmat-admin's picture

Hi Bertyy,

The quick answer: for the GMAT, only INTEGERS are considered even or odd.

Allowing fractions and decimals to be even or odd kind of blows up some of our trusted rules. For example, if we consider 0.25 and 0.5 to be odd, then 0.25 + 0.25 = 0.5 means that ODD + ODD = ODD

Of course this isn't to say that there's no side branch of mathematics where they consider what happens when we include fractions and decimals in the realm of evens and odds. But, for the purposes of the GMAT, we only consider INTEGERS.

Cheers,
Brent

I can deal with that. Not a problem. Lol!

https://gmatclub.com/forum/the-expression-ab-a-is-odd-when-the-a-and-b-are-integers-which-of-the-219205.html

I don't understand why E is not the answer.

I simplified the equation to if a(b+1) is odd then which must be even. Accordingly, it's not enough for b only to be even:

If b is even + odd = odd. That's all well and good but what about a? If a is even then no matter what happens you're going to get even. So the answer choice of b is not adequate.

It should ensure that both a and b are odd. The only answer that does this is E,

If a + b^2 must be even then,
a = odd
b = odd

odd + odd = even

Have I lost the plot or has there been a mathematical anomaly here?
gmat-admin's picture

Question link: https://gmatclub.com/forum/the-expression-ab-a-is-odd-when-the-a-and-b-a...

If (x)(y) = ODD, then we can be certain that x and y are both ODD.
In your approach, you noted that we can't make any conclusions about a, but we can.
If a(b+1) is ODD, then a is odd, and b+1 is odd.

Also, there's a problem when you made the following conclusion:
If a + b^2 must be even then,
a = odd
b = odd

If x + y = EVEN, then EITHER x and y are both ODD, OR x and y are both EVEN

Here's my full solution: https://gmatclub.com/forum/the-expression-ab-a-is-odd-when-the-a-and-b-a...

Does that help?

Cheers,
Brent

I came across this on the Quant flashcards provided by the gmatclub forums:

Is A + B + C = even or odd?

(1) A - C - B is even

Given the rule, EVEN +/- EVEN = EVEN and ODD +/- ODD = EVEN

I would have assumed that this is sufficient to prove that A - B - C is even.

What am I missing?
gmat-admin's picture

Statement 1 WOULD be sufficient IF we were told that A, B and C are INTEGERS.
For example, if A = 3.2, B = 0.5 and C = 0.7, then A - B - C = 2, which is even.
However, A + B + C = 4.4 which is not even.

Cheers,
Brent

https://gmatclub.com/forum/the-function-f-n-the-number-of-factors-of-n-if-p-and-q-73680.html

Hi Brent for this question, had we been asked "what is the value of Q"? Statement 2 would have sufficed right?

gmat-admin's picture

Question link: https://gmatclub.com/forum/the-function-f-n-the-number-of-factors-of-n-i...

If the target question were "What is the value of q?", statement 2 should would still be insufficient.

Statement 2: q is less than p
Consider these two cases:

CASE A: q = 2 and p = 3.
So, pq = (2)(3) = 6
Since 6 has four positive factors (1,2,3,6), we have satisfied the condition that f(pq) = 4.
In this case, the answer to the NEW target question is: q = 2

CASE B: q = 1 and p = 6.
Once again, pq = 6, which means we have satisfied the condition that f(pq) = 4.
In this case, the answer to the NEW target question is: q = 1

Since we can't answer the NEW target question with certainty, statement 2 is insufficient.

Does that help?

Cheers,
Brent

Hi Brent,

Could you please help me here?

https://gmatclub.com/forum/is-w-an-integer-218816.html

How is statement B sufficient? If I take w = 1.1 then 2w = 2.2 which is an even number but w is not an integer, and when I take w = 1 then 2w = 2. How can we conclude it be sufficient?

Thanks so much!
gmat-admin's picture

Question link: https://gmatclub.com/forum/is-w-an-integer-218816.html

Be careful; 2.2 is not considered even.
An even number is any INTEGER that's divisible by 2.

Cheers,
Brent

https://www.beatthegmat.com/if-x-is-an-integer-is-x-odd-t297304.html

I didn’t understand why A is the right answer and not D. In your solution for statement 2 being insufficient you have used the example of 4/3 but the answer, which is 1.3333 is not even an integer.

Could you please help me with this, thanks !
gmat-admin's picture

Question link: https://gmatclub.com/forum/is-x-an-odd-integer-141763.html

Target question: Is x odd?

It is given that x is an INTEGER.
Statement 2 tells us that x/3 is NOT an even integer.

Let's see what happens if x = 3
We can choose x = 3, because 3 is an integer (and we're told that x is an integer)
In this case, x/3 = 3/3 = 1
Since 1 is NOT an even integer, we can see that x = 3 satisfies BOTH the given information AND statement 2.

Let's see what happens if x = 3
We can choose x = 3, because 3 is an integer (and we're told that x is an integer)
In this case, x/3 = 3/3 = 1
Since 1 is NOT an even integer, we can see that x = 3 satisfies BOTH the given information AND statement 2.
If x = 3, then the answer to the target question is "YES, x is odd"

Now let's see what happens if 4 = 3
We can choose x = 4, because 4 is an integer (and we're told that x is an integer)
In this case, x/3 = 4/3 = 1.3333333.....
Is 1.3333333..... an even integer? (the answer to this question is either yes or no)
No, 1.3333333.....in NOT an even integer.
Since 1.3333333..... is NOT an even integer, we can see that x = 4 satisfies BOTH the given information AND statement 2.
If x = 4, then the answer to the target question is "NO, x is NOT odd"

Does that help?

Cheers, Brent

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