Number of distinct factors versus Number of factors

[maanda82]

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.

[Brent@GMATPrepNow]

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

Cheers,
Brent

[DavidG@VeritasPrep]

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.

[Brent@GMATPrepNow]

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