**Fact**: one of the most tested categories of concepts on the GMAT Quantitative section is Integer Properties

## What are the “properties of integers”?

Probably none of these are brand new to you —- in fact, you probably learned about all these in grade school. Here’s a list.

a) factors and multiples; GCF and LCM

b) quotient and remainder

c) even & odd

d) prime numbers

e) consecutive numbers and other consecutive sets (odds, evens, multiple of 5, etc.)

f) special properties of 1 (e.g. 1*(any number) = that number)

g) special properties of 0 (e.g. (any number) + 0 = that number, (any number)*0 = 0)

Again, nothing here is anything that is not covered in grade school, but the trick is: you have to have this all at your fingertips when you take the GMAT. These interrelated concepts lend themselves effortlessly to a myriad of PS & DS questions, and you need to handle them deftly with precision. See this post http://magoosh.com/gmat/2012/gmat-math-factors/ for tips about factors, multiples, GCF, and LCM. For more tips, see below.

## What are integers?

Integers are positive and negative whole numbers. They are the set:

{ . . . -3, -2, -1, 0, 1, 2, 3, . . .}

They go on forever in the positive and negative direction. They do not include fractions, decimals, and numbers like pi. One way for non-mathy folks to remember the integers: the word “integer” shares a root with the word “integrity” — both come from the Latin word for “whole, wholeness.” If I have integrity, there is a wholeness among my intention, my speech, and my actions; people who lack integrity say one thing and do another.

Big GMAT idea: if the GMAT makes a numerical statement (e.g. x < 3), do not assume x is an integer unless that is specified. That’s one of the biggest DS traps, assuming the only possibilities are integers when there are many more possibilities allowed. Here’s a trippy advanced math idea: the infinity of non-integers is infinitely bigger than the infinity of integers. (Read http://en.wikipedia.org/wiki/Infinity if you are up for an adventure learning about infinity — well beyond what you need to know for the GMAT).

## Even and Odd

First of all, here are three addition rules:

1) (even) + (even) = (even)

2) (odd) + (odd) = (even)

3) (even) + (odd) = (odd)

Those also work if the addition sign is changed to a subtraction sign.

Now, three multiplication rules

1) (even)*(even) = (even)

2) (odd)*(odd) = (odd)

3) (even)*(odd) = (even)

(These rules are ** not** the same if the multiplication sign is changed to division!) If you have trouble remembering these six rules, you can always use even = 2 and odd = 3 to remind yourself. (Yes, 1 is also odd, but I recommend not using that as a test number only because it has so many special properties.)

Keep in mind: zero is an even number. Keep in mind, also: negative numbers can be even and odd, just like their positive compatriots. Fractions and non-integers cannot be even or odd: it’s exclusively an integer property.

## Prime

Every number has 1 as a factor. Every number has itself as a factor. A number is prime if it has only those two factors, i.e., no factors other than 1 and itself. Only positive integers are said to be prime; we do not apply the distinction “prime” or “not prime” to negative integers, zero, or to non-integers.

Here is a list of the first few primes:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, . . .

The primes go on forever in an irregular pattern, the nature of which involves the hardest unanswered question in math today, the Riemann Hypothesis — again, well beyond what you need to know for the GMAT. It would be good to memorize that list of the first ten prime numbers: that will help you a lot on the GMAT. Notice that, for a variety of reasons with which we need not concern ourselves here, 1 is not a prime number. Notice, also: 2 is the only even prime number: all other even numbers are divisible by 2. That’s a very handy distinction, especially in GMAT DS: 2 is the only even prime number.

See the factors post for more information about prime numbers.

## Know Them Cold

Probably there’s nothing brand new in this post. Probably you have at least a dim memory, if not a perfectly clear understanding, of everything here. Whatever here is rusty, whatever is less than perfectly fluent, you need to practice until you know it cold. The GMAT is relentless in asking about these properties, and if you can nail them every time, you will be well on your way to a stunningly successful GMAT Quantitative section.

Here are a couple of practice questions:

1) http://gmat.magoosh.com/questions/880

2) http://gmat.magoosh.com/questions/317

### Most Popular Resources

I am wondering when an Integer MUST be positive, I think I may have misinterpreted some previous videos and want to confirm I thought factors and multiples could be negative as well.. But the video “DS with integer contraints” says otherwise.

Hi Stephen! You bring up some excellent questions! Technically, factors and multiples may be negative or positive. For example, negative integers (like -2 and -3) can technically be factors of positive even integers (like 6). However, on the GMAT, we suggest that you just assume you’re dealing with the positive factors of an integer unless you’re instructed to do otherwise.

Also, as Mike notes in the lesson you’ve asked about, you may run into some word problems where you’ll have to determine the factors of a number of concrete physical items (like boxes, employees, and so on). For those problems, you have to get an answer that’s a positive integer, since there can’t be a negative number of boxes or anything like that. So again, in those sorts of problems, it’s pretty useful to assume that the integers you’re factoring are all positive. 🙂

In general, the questions will usually specific “positive” factors or divisor, but if they don’t, negative factors are fair game.

I hope this clears up your doubts!

Hello there. Congrats on putting up an excellent blog. But I’ve question that has bothered me for a while now. When counting multiples for a number, do we consider negative numbers too? I’ve tried searching my answer at various sites and blogs and there remains a confusion. It’ll be really helpful if you can solve this mystery for me. Thanks. Y.

In the strict rules of math, negative numbers count as multiples. However, on the GMAT exam, when a math question refers to “multiples,” it is only referring to positive multiples, unless the question specifically says that negative multiples are also included.

Hey Mike im trying to hunt for a list with all the topics for Quant (like broken down, not broad topics). I’ll be starting GMAT preparations tomorrow onwards so it’ll really help.

Thanks,

Candice

Candice: first of all, this post: http://magoosh.com/gmat/2012/breakdown-of-gmat-quant-concepts-by-frequency/ lists all the GMAT math topics by frequency. Notice that the breakdown follows precisely the outline of the “Math Review” section of the OG (for example, OG 13 pp. 107-146). The idea is — you can look at the blog to see the most frequent topics, and if there’s any term for which you would like more clarification, it’s right there in the OG. Does that make sense?

Mike 🙂

Well yeah its exactly what i wanted, Cheers!

Candice: I’m glad I could help. Best of luck to you!

Mike 🙂

Hi mike

can you explain this

How many different positive integers exist between 106 and 107, the sum of whose digits is equal to 2?

A. 6

B. 7

C. 5

D. 8

E. 18

for the above question it is 10^6 and 10^7

Dear Sony — the question as phrased makes no sense, because 106 and 107 are right next to each other as integers and there are absolutely no integers between them. I searched the web, and found a slight variant that makes much more sense:

“How many different positive integers exist between 10^6 and 10^7, the sum of whose digits is equal to 2?”

Those are integers between 1,000,000 and 10,000,000 whose digits sum to 2. Well, there are two choices — two 1’s, and the rest zeros; or one 2, and the rest zero. Since zero can’t be the first digit, the only possibility in the second case is 2,000,000. In the first case, one of the 1’s has to be the first digit, and the other can go in any of six remaining places: 1,000,001 & 1,000,010 & 1,000,100 & 1,001,000 &1,010,000 and 1,100,000. Six cases with two 1’s and one case with one 2, for a total of seven cases — answer = B.

I assume the 10^6, not 106, question is the question you were asking. If I did not answer the question you intended, or if you have any further questions, please let me know.

Mike 🙂

Sony

Your clarification comment just came through, so I was answering the correct question. Is everything clear now?

Mike 🙂

Thank u very much Mike

You are quite welcome. Best of luck to you, and let us know if you have any further questions. Mike 🙂

6 + 1 is not equal to 8!

Thank you: In the text, I came up with the correct answer of 7, but for whatever reason, I put the incorrect answer choice. It’s now corrected.

Mike 🙂

good work, remembering these things are crucial in DS. I had a doubt regarding OG, I found OG math is very easy and if we r perfect with those models, what’s the score can we expect,all i want to know is min marks gaurenteed by doing og in general

Dear veeramani,

If you find the OG Quant questions easy *and* can regularly get just about all of them correct, then I would think that your Quant score would be in the elite 50+ region. It’s true, the concepts are not all that difficult, for someone who has a math background, but the key is not to let a single exception, a single distinction, evade your notice. Finding them easy is one thing, and getting them almost all correct is another. Does this distinction make sense?

Mike 🙂

The following points of the post look simple but are very important and may make the GMAT-taker bowled if he is not thorough about them:

“do not assume x is an integer unless that is specified.”

“Keep in mind: zero is an even number. Keep in mind, also: negative numbers can be even and odd, just like their positive compatriots”

“Only positive integers are said to be prime”

A Great work, Mike.

Thank you for your kind words. Best of luck to you, and please let us know if we can be of any further assistance.

Mike 🙂

Hi Mike,

I have doubt in the question below:

http://gmat.magoosh.com/questions/864

Can you please explain how did you conclude that Statement B is also true. Why did we not consider n=1?

Thanks

Yes, there’s an error in that question. As I explained in p.m. on BTG, you’re correct — n = 1 is possibility for Statement #2, so D is not the answer.

Mike 🙂

Hello Magoosh team,

I have a doubt in DS type of questions.

If the answer choice is D, then should the result/answer obtained in 1st statement always be the same as that obtained in 2nd statement.

I am attaching a problem that I got outside, where I can clear my above statement.

If n is an integer, then n is divisible by how many positive integers?

(1) n and 2^4 are divisible by the same number of positive numbers.

(2) n is the product of 2 and an odd prime number.

Answer is D . Though we get answer in both the statements, they are different in the one case the answer is 5 and in the second one it is 4. Shouldn’t we get the same answer from both the statements for it to be D?

I am sorry if I have posted it in a wrong thread. But request your detailed reply that will help me a lot.

Regards,

Harsha

Harsha: There are many folks out there preparing GMAT questions, and not all of them are up to the same quality. I don’t know the source of this DS question, but this is typical of DS questions from a low quality source. In real GMAT questions, and in questions from all high quality GMAT test prep sources, like Magoosh, you will find in a “D” answer, both statements always lead to the same numerical answer. The only time that doesn’t happen is if the source of your questions does not provide high quality. I would strongly recommend not using such sources, including wherever you found that question. Does that make sense?

Mike 🙂

Although these are just basic rules, I often forget some of these guidelines. These tips are really helpful, thank you!

Lyn: I’m glad to hear that this was helpful. Thank you for telling us. Best of luck to you, and let us know if we can support you any further.

Mike 🙂