Finding Factors of Large Nums

Question

Hi everyone, I'm beginning my GMAT Quant journey with the MGMAT Number Properties guide and the Kaplan Math Workbook (6th ed.). I understand the concept of factoring, finding GCFs, etc, but I'm wondering if there is a quick/tried-and-true strategy for finding factors of large numbers without having to create a chart, which seems too time consuming. Thanks in advance for the help!

firasath · Answer

Not sure what your "chart" method is but an easy method to remember and use is to find the prime factorization and then expand the permutations of the exponents:

example:

196
 
196 = 2^2*7^2 (prime factorization)

To determine how many factors there are, just look at each power, add one, and multiply together.

(2 + 1)(2 + 1)
(3)(3)
9 factors

now work through the permutations of the exponents starting with 0:

2^0*7^0 = 1*1 = 1
2^0*7^1 = 1*7 = 7
2^0*7^2 = 1*49= 49

2^1*7^0 = 2*1=2
2^1*7^1 = 2*7=14
2^1*7^2 = 2*49 = 98

2^2*7^0 = 4*1=4
2^2*7^1 = 4*7=28
2^2*7^2 = 4*49 = 196

thus, the factors of 196 are 1, 2, 4, 7, 14, 28, 49, 98, 196

amenewyork · Answer

Great, thank you for that method!

AshForGMAT · Answer

Yes the above method is the shortest method that i have come across...

Please share if you have come across anything shorter !

tarun · Answer

I agree. This is the shortest method.

fruit · Answer

Thanks for this method.
And what about the number 483?
We get three prime numbers 7, 23 and 3. What to do next?

Bunuel · Answer

Finding the Number of Factors of an Integer

First make prime factorization of an integer  n=a^p*b^q*c^r , where  a ,  b , and  c  are prime factors of  n  and  p ,  q , and  r  are their powers.
 
The number of factors of  n  will be expressed by the formula  (p+1)(q+1)(r+1) .  NOTE:  this will include 1 and n itself.

Example:  Finding the number of all factors of 450:  450=2^1*3^2*5^2

Total number of factors of 450 including 1 and 450 itself is  (1+1)*(2+1)*(2+1)=2*3*3=18  factors.

483=3^1*7^1*23^1  --> number of factors of 483 is:  (1+1)(1+1)(1+1)=8 .

For more on number properties see the Number Theory chapter of Math Book (link in my signature).

Hope it helps.

fruit · Answer

I meant how to find all 8 factors? As you did in your example:

Does GMAT check this knowledge?

firasath · Answer

3^0*7^0*23^0 = 1
3^0*7^0*23^1 = 23

3^0*7^1*23^0 = 7
3^0*7^1*23^1 = 161

3^1*7^0*23^0 = 3
3^1*7^0*23^1 = 69

3^1*7^1*23^0 = 21
3^1*7^1*23^1 = 483

your factors are: 1,3,7,21,23,69,161,483

Exellon · Answer

How do you quickly determine a,b, c and p, q, r in the typical n=a^p*b^q*c^r formula?

Seems like guesswork to me...

Finding Factors of Large Nums

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