## is there a formula to calculate the number of factors?

Questions about the world of GMAT Math from other sources and general math related questions.
gt7er

### is there a formula to calculate the number of factors?

what is the number of factors for 441? It breaks down into 3, 3, 7, 7 ...how do you find the total number of factors? what's the formula?
H_to_750

### no formula

There is no formula you just take a look at how many different combinations there are of prime factors

So in this case

WE have 3, 7, (7 x 7 = 49), (7 x 3 = 21), (3 x 3 = 9) (9 x 7 = 63), (49 x 3 = 147) , 441, and 1

So for 441 we have total of 9 factors. Just always remember that 1 and the number it self is also a factor.
David Pollack

### A nice formula for the number of factors

Sure, there is! It just depends on how many prime factors your number has, and how many times each one appears.

Suppose your number factors as a product p^a * q^b * .... *r^k. Then to get any factor you want you should take _up to_ a copies of p, _up to_ b copies of q, etc. and multiply them all together. The number of ways you can choose _up to_ a copies of p is (a+1) since you could choose 0 copies, 1 copy, 2 copies, ..., a copies. Likewise there are (b+1) ways to choose how many q's to include, etc.

So the number of factors for your number would be (a+1)*(b+1)*...(k+1).

For example, if your number is 441 = 3^2 * 7^2 then the number of factors is (2+1)*(2+1)=3*3=9.

One more example: if your number is 360 = 8*9*5 = 2^3 * 3^2 * 5^1 then the number of factors is (3+1)*(2+1)*(1+1)=24.
RonPurewal
ManhattanGMAT Staff

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Joined: Tue Aug 14, 2007 8:23 am

### Re: A nice formula for the number of factors

David Pollack wrote:Sure, there is! It just depends on how many prime factors your number has, and how many times each one appears.

Suppose your number factors as a product p^a * q^b * .... *r^k. Then to get any factor you want you should take _up to_ a copies of p, _up to_ b copies of q, etc. and multiply them all together. The number of ways you can choose _up to_ a copies of p is (a+1) since you could choose 0 copies, 1 copy, 2 copies, ..., a copies. Likewise there are (b+1) ways to choose how many q's to include, etc.

So the number of factors for your number would be (a+1)*(b+1)*...(k+1).

For example, if your number is 441 = 3^2 * 7^2 then the number of factors is (2+1)*(2+1)=3*3=9.

One more example: if your number is 360 = 8*9*5 = 2^3 * 3^2 * 5^1 then the number of factors is (3+1)*(2+1)*(1+1)=24.

this is just awesome. very well done.

i have some unsolicited advice for you:
step 1) drop whatever you're doing
step 2) write a textbook
step 3) resume whatever you're doing

thank you for saving us the time required to answer this ourselves. :)
Guest

### Re: A nice formula for the number of factors

RPurewal wrote:
David Pollack wrote:Sure, there is! It just depends on how many prime factors your number has, and how many times each one appears.

Suppose your number factors as a product p^a * q^b * .... *r^k. Then to get any factor you want you should take _up to_ a copies of p, _up to_ b copies of q, etc. and multiply them all together. The number of ways you can choose _up to_ a copies of p is (a+1) since you could choose 0 copies, 1 copy, 2 copies, ..., a copies. Likewise there are (b+1) ways to choose how many q's to include, etc.

So the number of factors for your number would be (a+1)*(b+1)*...(k+1).

For example, if your number is 441 = 3^2 * 7^2 then the number of factors is (2+1)*(2+1)=3*3=9.

One more example: if your number is 360 = 8*9*5 = 2^3 * 3^2 * 5^1 then the number of factors is (3+1)*(2+1)*(1+1)=24.

this is just awesome. very well done.

i have some unsolicited advice for you:
step 1) drop whatever you're doing
step 2) write a textbook
step 3) resume whatever you're doing

thank you for saving us the time required to answer this ourselves. :)

That explanation less the examples read more like directions to fix a pinball machine in chinese
Dlew 686

Yeah I agree.
iil-london

### Prime factors ... quickest way ?

Hi there ...

what is the quickest way to find the prime factors for a number ?
How would you find the prime factors of 3841, for example ?
Does anyone know of a quick way to do it.

Thanks for the help. Appreciate it.
upa

### Re: Prime factors ... quickest way ?

iil-london wrote:Hi there ...

what is the quickest way to find the prime factors for a number ?
How would you find the prime factors of 3841, for example ?
Does anyone know of a quick way to do it.

Thanks for the help. Appreciate it.

I guess there is not. If anybody has I appreciate him/her for his/her brilliant idea.

Just factorize it. I know it is not easy and doesnot have factors 2, 3, 5, 7, 11, 13, 17, and 19. but 23 divides it.

3841 = 23 x 167

i guess 167 is also a prime cuz if it were not a prime it has to be divided by 2, 3, 5, 7, or son. none of them divides 167, so its a prime.
this is a tough one.
StaceyKoprince
ManhattanGMAT Staff

Posts: 8450
Joined: Wed Oct 19, 2005 9:05 am
Location: San Francisco

David - I agree, nice process. And for those of you who weren't quite sure how to follow the theory - don't worry about it! Just follow the examples :)

I don't know of a shortcut to find the complete set of prime factors for a number very quickly. I'm guessing there isn't one because one of the weird things about primes is that they pretty much defy our attempts to reduce things to typical patterns or formulas.
Stacey Koprince
Instructor
Director of Online Community
ManhattanGMAT
Guest

### finding primes

Not exactly a shortcut to finding the primes, but to help determine a large prime. Take the square root of the number. You only have to try the primes up through the one just larger than the square root.

In other words, sqrt 167 = 12.9... so you only have to try dividing it by the prime numbers through 13. Since none of those work, it is prime.
JonathanSchneider
ManhattanGMAT Staff

Posts: 477
Joined: Wed Dec 12, 2007 5:40 am
Location: Durham, NC

### Re: is there a formula to calculate the number of factors?

Good point - glad you added that!
zxtonizx
Course Students

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Joined: Tue Nov 22, 2011 1:45 am

### Re: is there a formula to calculate the number of factors?

Cannot tell you the time David Pollack's shortcut has saved. Thank you sir!
tim
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### Re: is there a formula to calculate the number of factors?

cool
Tim Sanders
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jp.jprasanna
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### Re: is there a formula to calculate the number of factors?

Hi - I undestand the below method, which gives me total factors, but is there a way I can find the numbers itself? Apart from listing them out!

One more example: if your number is 360 = 8*9*5 = 2^3 * 3^2 * 5^1 then the number of factors is (3+1)*(2+1)*(1+1)=24

Cheers
tim
ManhattanGMAT Staff

Posts: 5672
Joined: Tue Sep 11, 2007 9:08 am
Location: Southwest Airlines, seat 21C

### Re: is there a formula to calculate the number of factors?

no, you just have to list them out..
Tim Sanders
Manhattan GMAT Instructor

Follow this link for some important tips to get the most out of your forum experience:
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