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### How Many Factors?

```
Date: 7/14/96 at 19:6:19
From: PANI BUNDY
Subject: How Many Factors?

Dear Dr.Math

Is there a formula to find the number of factors for a number?

For example:   The factors of 12 are 12,6,4,3,2,1
...so there are 6 factors of 12.

Do you know what the formula is to find the number of factors for any
number?

Thank you in advance,

Pani Bundy
```

```
Date: 7/15/96 at 7:37:39
From: Doctor Anthony
Subject: Re: How Many Factors?

To find the number of factors you must first express the number in
prime factors.  To take your example, 12 = 2^2 * 3

Now the factor 2 can appear 0, 1 or 2 times (i.e. in 3 different ways)
and the factor 3 can occur 0 or 1 times (i.e. in 2 different ways).
So the total number of ways the factors can occur is 3 * 2 = 6 ways.
So there are 6 possible factors of 12 (including 1 and 12)

The general method is to express the number in prime factors, e.g.

N = p1^r1 * p2^r2 * p3^r3

Then p1 could occur in r1 + 1 ways
p2 could occur in r2 + 1 ways
p3 could occur in r3 + 1 ways.

Total number of factors = (r1 + 1)(r2 + 1)(r3 + 1)

Example.  Find the number of possible factors of 144

144 = 2^4 * 3^2

The factor 2 could appear in 5 ways, and the factor 3 in 3 ways, so
the total number of factors = 5 * 3 = 15 factors.

-Doctor Anthony,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```
Associated Topics:
High School Discrete Mathematics

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