# Lesson: Even and Odd Integers

## Comment on Even and Odd Integers

### 0 is neither even nor odd.

0 is neither even nor odd. ### Zero is even.

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

zero is neither positive nor negative That's true

### appears this lesson (and the

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 ### I don't have such a list.

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,

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 ### Hi Bertyy,

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

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

### https://gmatclub.com/forum

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? 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

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? ### Statement 1 WOULD be

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

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? 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,

Hi Brent,

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! 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

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. 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

### https://gmatclub.com/forum/if

https://gmatclub.com/forum/if-n-and-p-are-positive-integers-what-is-the-ratio-of-n-to-p-1-n-241219.html

IMPORTANT: *If p and n are integers, then p-n and p+n will also be integers.
Also, if p+n is ODD then p-n must also be ODD
Likewise, if p+n is EVEN then p-n must also be EVEN*

I thought like this:
As RHS is 28,so LHS Will (both even) or (one even)(one odd).

Would you please correct me in this regards?
Thanks. That's a good idea, but the second case you suggest, "(one even)(one odd)" isn't possible for this given question.

Here are some properties we need to know about odds and evens:
#1. ODD +/- ODD = EVEN
#2. ODD +/- EVEN = ODD
#3. EVEN +/- EVEN = EVEN

Statement 2 tells us that (p - n)(p + n) = 28

If (p + n) is ODD, then Property #2, tells us that one of the values (p or n) is even, and the other value is odd.
If one value is even and the other value is odd, then it must also be true that (p - n) is ODD (according to Property #2).

So it's impossible to have a situation where (p + n) is ODD, yet (p + n) is EVEN.

Does that help?

### The function f(n) = the

The function f(n) = the number of factors of n. If p and q are positive integers and f(pq) = 4, what is the value of p?

(1) p + q is an odd integer

(2) q is less than p

Statement 2: q is less than p
From the given information, we know that p and q are both prime numbers.

Didn't understand this part. How Both are primes? Good question! I have no idea why I mentioned that in my solution.

It COULD be the case that p and q are both prime numbers.
For example, if q = 2 and p = 7, then pq = 14, and the divisors of 14 are: 2, 7, 1 and 14 (4 divisors)

HOWEVER, it could also be the case that p and q are NOT both prime numbers.
For example, if q = 2 and p = 4, then pq = 8, and the divisors of 8 are: 1, 2, 4, and 8 (4 divisors)

I have since removed that line from my solution.

### Hi Brent,

Hi Brent,
Pls explain this

If x, y, and z are positive numbers, is x - y < z?

(1) x - y < z
(2) xy < z^2 ### Here's my full solution:

Here's my full solution: https://gmatclub.com/forum/if-x-y-and-z-are-positive-numbers-is-x-y-z-30...

Cheers,
Brent
PS: In the future, to speed things up, you can just paste the link to the question.

### Hi Brent,

Hi Brent,

If the sum of three integers is even, is the product of the three integers a multiple of 4 ?

(1) All three integers are equal.

(2) All three integers are even.

Can we consider 0 in this case? Yes, 0 is a multiple of 4, AND 0 is also even.

### Hi Brent,

Hi Brent,
Is the integer n even?

(1) n – 5 is an odd integer.
(2) n/5 is an even integer.
Pls tell me whether I comprehend in the right direction for the statement 2 i.e. Even/Odd yields an integer it must be an even integer hence N is even. Am I right? That's correct. Here's the property:
If P is an EVEN integer, Q is an ODD integer, and P/Q = integer N, then we can conclude that N is EVEN.

Cheers, Brent

### Hi Brent ,

Hi Brent ,
What should be my approach to this problem

https://gmatclub.com/forum/if-u-is-an-odd-number-and-v-and-w-are-different-integers-which-of-the-223767.html For this question, I'd probably plug in some values and eliminate answer choices.
Here's a solution that uses this approach: https://gmatclub.com/forum/if-u-is-an-odd-number-and-v-and-w-are-differe...

### Hi Brent,

Hi Brent,

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

Could you please explain why the answer is B, given that even/even = non-integer, even or odd? It would be better if the statements used "integer" instead of "number" (official GMAT questions involving evens and odd always use the term "integer")

The idea here is that only integers are considered even or odd.
So statement 1 is telling us that 2w is an even integer.
There's a property that says: All even integers can be expressed as 2k, where k is an integer.
So, if 2w is an even integer, we can say: 2w = 2k (for some integer k)
Divide both sides by 2 to get: w = k (for some integer k)

Does that help?

### Thanks, that makes a lot more

Thanks, that makes a lot more sense!