Number Properties, Part I: Factors, Multiples, and Prime Factorization
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Brian is a GMAT Instructor living in Washington, D.C. Click here to read more articles from Kaplan and to learn more about Kaplan's GMAT classes. |
Studying for the quantitative section of the GMAT requires remembering a few concepts from high school – concepts that you never thought you would need again. Number properties (or the properties of numbers) are probably one of those topics.
Each number has a distinct set of properties attributed to it. For example, multiplying an odd number by an odd number always results in another odd number. For the GMAT, there is no need to go into any proofs – it is sufficient to remember some of the key components of the number properties and work from there. Today, let’s talk about the attributes associated with Multiples and Factors – two concepts that beginning GMAT students often confuse.
Before we jump into each of these areas, let’s take a high-level look at the difference between factors and multiples. Let’s pick the number 8.
What are the Factors of 8? 1, 2, 4, 8
What are the Multiples of 8? 8, 16, 24, 32, etc.
The takeaway here is that Factors are the small guys – anything below the number in question that can be multiplied by another number to arrive at the destination “8”. Multiples are the BIG guys – anything above the number in question that is multiplied by another number. Memorize this distinction – you don’t want to be sitting there on test day trying to recall Mr. Watson’s class regarding factors and multiples just to do a problem. Let’s look at these a little further…
Factors
It is important to be able to ‘translate’ Quantitative word problems into algebra on test day. Factors play an important part of this translation: X is a FACTOR of Y – meaning, x multiplied by another integer, will equal Y. Or, in algebraic terms: X * N = Y.
Multiples
In the same vein, translating GMAT ‘multiples’ questions are also very important: X is a MULTIPLE of Y – meaning, Y * N = X. Notice the difference between the two algebraic expressions? A pretty big difference for Problem Solving questions; however, the same data points would be sufficient for a Data Sufficiency questions.
Last concept for today: Prime Factorization
Prime Factorization is another important concept on the GMAT. For the purposes of this blog post – focusing on Number Properties – Prime Factorization focuses on the all the Prime Factors that make up a GMAT number. Let’s look at the number 30. 30 has the following factors: 1, 2, 3, 5, 6, 10, 15, 30. Since Prime numbers are numbers that can not be divided by any number except for itself and 1, we see the prime factors of 30 are the following: 2, 3, 5. Every other factor can be divided further by those three prime numbers (i.e. 6 can be factorized as 2 & 3). Thus, the GMAT may ask: “How many prime factors of 30 are there?” The answer is three.
Factors and Multiples are some of the Number Properties concepts you’ll see on test day. Don’t let relatively basic concepts throw you for a loop – the GMAT isn’t testing your ability to Factor 5,312 – they are testing your ability to break down seeming complex situations into straight-forward mathematical arguments.
aks on August 19th, 2009 at 12:23 am
In the Prime Factors of 30, why is "1" not included?
Brad on August 19th, 2009 at 12:37 am
The number 1 isn't listed as a prime factor of 30, because 1 isn't prime. Brian has used the simplified definition of prime numbers: "Prime numbers are numbers that can not be divided by any number except for itself and 1."
I don't want to speak for him, but this is probably because it's a little easier to remember than the formal definition of a prime: a natural number which has exactly two distinct natural number divisors - 1 and itself. 1 does not have two distinct natural divisors (it has a sole natural divisor, itself) and doesn't qualify.
Long story short, 1 is not a prime number. Always remember 2 is the smallest and only even prime.
Excellent article by Brian, who, by the way, is a fantastic instructor.
mymsfstudy on August 19th, 2009 at 10:30 pm
"1" is not a prime number
The least prime number is "2", and it's also the only prime that's not an odd number
aks on August 19th, 2009 at 9:23 am
Thanks for the clarification.
Sven Kämmerer on August 19th, 2009 at 11:20 am
Whats up with zero being a multiple of every number? That's true, isn't it?
Andrew at Kaplan on August 20th, 2009 at 2:04 pm
Since a factor * N = multiple, zero is a multiple of every number. Any factor * 0 = 0.
Another way of thinking about it is: zero divided by any number is zero with no remainder. So any number is a factor of zero (that's why zero is considered even - 2 is a factor of it), and zero is a multiple of any number.
njb on August 19th, 2009 at 6:01 pm
36 is NOT a multiple of 8!!!
Eric Bahn on August 19th, 2009 at 8:21 pm
You are very correct. Just made an update to this post.
gmat_guy on August 19th, 2009 at 10:30 pm
the number 0(zero) is also a multiple of every integer, which shudnot be forgotton. one of the questn in kaplan gmat involves this fact..
is x a prime?
i) x<15
ii) x-2 is a multiple of 5
Answer: E, 7-2=5, 2-2=0, both are multiples of 5, so A&B both combined not sufficient, as x can be 7 or 2.
sa on August 20th, 2009 at 5:34 am
Actually, if x-2 is a multiple of 5, then combining 1 & 2, we get (x-2)/5 is N (integer). Therefore x-2 = 5N, so x-2 can be 5, or 10 (since x<15).
Thus we get x=7 or x=12, therefore E. So not sure if we should consider 0 as multiple of every integer.
gmat_guy on August 19th, 2009 at 11:42 pm
the question above should be read as:
what is the value of x?
sorry for the typo.
gmat_guy on August 19th, 2009 at 11:44 pm
the question above shud be read as:
what is the value of x?
sorry for the typo.
Mia on August 19th, 2009 at 11:47 pm
8 is both a factor and multiple of 8.So for every integer n there is 1 number which is both its factor and its multiple at the same time(which is n). Correct?..
gmat_guy on August 20th, 2009 at 7:27 pm
this is absolutely correct. a number is simultaneously, both its factor and its multiple.
gmat_guy on August 20th, 2009 at 7:24 pm
0 is a multiple of every integer bcoz,
8/8=1(integer), 16/8=2(integer), 24/8=3(integer)....so, 8,16,24.....etc are multiples of 8.
Now, 0/8=0(integer), hence it can be shown that 0 is also a multiple of every integer.
M on September 1st, 2009 at 12:26 am
question:
is P a prime factor of 168?
i) P is afactor of 14
ii) P is a factor of 12
data sufficiency
jitendra on September 7th, 2009 at 7:08 am
From i) above, P can be 2,7 or 14
From ii) above, P can be 2,3,6 or 12
combining both, P = 2 so option (C)..is it correct?
briantime on October 18th, 2009 at 9:50 am
From i) above, P can be 2,7 or 14
From ii) above, P can be 2,3,6 or 12
combining both, P = 2 so option (C)..is it correct?
----
I think so, but you forgot the factor "4" in Statement II.
g-train on November 9th, 2009 at 4:18 pm
I) the factors of 14 are (1,2,7,14), so A and D are out.
2) the factors of 12 are (1,2,3,4,6,12), so B is out.
combining both statements, we see that 14 and 12 share prime factor 2, but also, they share non-prime 1.
this makes the answer choice d, right? we can't determine if the answer is 1 or 2.
anyone? anyone?
Bill Bartmann on September 18th, 2009 at 4:14 pm
Great site...keep up the good work. I read a lot of blogs on a daily basis and for the most part, people lack substance but, I just wanted to make a quick comment to say I'm glad I found your blog. Thanks,
A definite great read..
-Bill-Bartmann
sumnut on November 19th, 2009 at 6:11 am
Here's a question I can't figure out: The prime factorization of a natural number n can be written as n=pr^2 where p and r are distinct prime numbers. How many factors does n have, including one and itself?