Number of distinct factors versus Number of factors

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maanda82: Newbie | Next Rank: 10 Posts; Posts: 2; Joined: Sat Oct 18, 2014 3:20 am

by maanda82 » Tue Apr 28, 2015 5:14 am

I came across something I have found confusing regarding counting the number of factors.

I seen a question asking for the 'number of factors' of a square. One of the answers gives the factors for 36 as (1, 36) (2, 18) (3,12) (4, 9) (6, 6). It then counts them as 9.

My question is when counting factors should I ignore duplicates? The only time I count duplicates is when the question asks for distinct factors?

Thank you in advance.

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Source: Beat The GMAT — Problem Solving | Tue Apr 28, 2015 5:14 am

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by Brent@GMATPrepNow » Tue Apr 28, 2015 9:38 am

Yes, ignore the duplicates.
The POSITIVE factors/divisors of 36 are: 1, 2, 3, 4, 6, 9, 12, 18, 36
NOTE: the GMAT typically restricts the question to POSITIVE divisors. However, if there is no restriction, 36 actually has 18 divisors: 1, -1, 2, -2, 3, -3, 4, -4, 6, -6, 9, -9, 12, -12, 18, -18, 36, -36

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DavidG@VeritasPrep: Legendary Member; Posts: 2663; Joined: Wed Jan 14, 2015 8:25 amFollowed by: 128 members; Location: Boston, MA; Thanked: 1153 times; GMAT Score:770

by DavidG@VeritasPrep » Fri May 01, 2015 3:52 am

A nice shortcut for finding the number of divisors of any number: (we'll illustrate with 36.)

First, take the prime factorization: 36 = 2^2 * 3^2

Now take the exponent of each prime base and add one. Then multiply those numbers. So we'll get (2+1)(2+1) = 3*3 = 9 factors.

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Brent@GMATPrepNow: GMAT Instructor; Posts: 16207; Joined: Mon Dec 08, 2008 6:26 pmFollowed by: 1268 members; Location: Vancouver, BC; Thanked: 5254 times; GMAT Score:770

by Brent@GMATPrepNow » Fri May 01, 2015 8:15 am

Here's a formal definition of the shortcut that David is talking about.

If N = (p^a)(q^b)(r^c)..., where p, q, r,...(etc.) are prime numbers, then the total number of POSITIVE divisors of N is equal to (a+1)(b+1)(c+1)...

Example: 14000 = (2^4)(5^3)(7^1)
So, the number of positive divisors of 14000 = (4+1)(3+1)(1+1) = (5)(4)(2) = 40

Cheers,
Brent

Brent Hanneson - Creator of GMATPrepNow.com

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