[Mental Math]

If a number can be expressed as a product of two whole numbers, then the whole numbers are called factors of that number.
In other words, A factor is a whole number which divides exactly into a whole number, leaving no remainder. For example, 13 is a factor of 52 because 13 divides exactly into 52 (52 ÷ 13 = 4 leaving no remainder).
52 = 1 x 52 = 2 x 26 = 4 x 13
So, the complete list of factors of 52 is: 1, 2, 4, 13, 26, and 52 (all these divide exactly into 52).
The simple technique to find the number of factors of a given number is to express the number as a product of powers of prime numbers or prime factors.
To illustrate let's find the numbers of factors of our example 52. Note that, 52 can be expressed as 4 x 13 = 2² x 13 So, the prime factors of 52 are 2 and 13.
Now, increment the power of each of the prime numbers by 1 and multiply the result. In this case it will be (2 + 1) x (1 + 1) = 3 x 2 = 6 (power of 2 is 2 and power of 13 is 1) Therefore, there will 6 factors including 1 and 52.
Also note that, all numbers have a factor of 1 since 1 multiplied by any number equals that number. All numbers can be divided by themselves to produce the number 1. Therefore, we normally ignore 1 and the number itself as useful factors.
So, excluding, these two numbers, you will have (6 – 2) = 4 factors. To be certain the factors are: 2, 4, 13 and 26.

To further illustrate let's find the numbers of factors of 48.
48 can be written as 16 x 3 = 2⁴ x 3 So, the prime factors of 48 are 2 and 3. Now, increment the power of each of the prime numbers by 1 and multiply the result. In this case it will be (4 + 1) x (1 + 1) = 5 x 2 = 10 (the power of 2 is 4 and the power of 3 is 1) Therefore, there will be 10 factors including 1 and 48.
Excluding, these two numbers, you will have (10 – 2) = 8 factors. And the factors are: 2, 3, 4, 6, 8, 12, 16 and 24.

This post has been edited by Mental Math: 12 February 2012 - 10:02 AM

[DrRocket]

Mental Math, on 12 February 2012 - 10:01 AM, said:

If a number can be expressed as a product of two whole numbers, then the whole numbers are called factors of that number.
In other words, A factor is a whole number which divides exactly into a whole number, leaving no remainder. For example, 13 is a factor of 52 because 13 divides exactly into 52 (52 ÷ 13 = 4 leaving no remainder).
52 = 1 x 52 = 2 x 26 = 4 x 13
So, the complete list of factors of 52 is: 1, 2, 4, 13, 26, and 52 (all these divide exactly into 52).
The simple technique to find the number of factors of a given number is to express the number as a product of powers of prime numbers or prime factors.
To illustrate let's find the numbers of factors of our example 52. Note that, 52 can be expressed as 4 x 13 = 2² x 13 So, the prime factors of 52 are 2 and 13.
Now, increment the power of each of the prime numbers by 1 and multiply the result. In this case it will be (2 + 1) x (1 + 1) = 3 x 2 = 6 (power of 2 is 2 and power of 13 is 1) Therefore, there will 6 factors including 1 and 52.
Also note that, all numbers have a factor of 1 since 1 multiplied by any number equals that number. All numbers can be divided by themselves to produce the number 1. Therefore, we normally ignore 1 and the number itself as useful factors.
So, excluding, these two numbers, you will have (6 – 2) = 4 factors. To be certain the factors are: 2, 4, 13 and 26.

To further illustrate let's find the numbers of factors of 48.
48 can be written as 16 x 3 = 2⁴ x 3 So, the prime factors of 48 are 2 and 3. Now, increment the power of each of the prime numbers by 1 and multiply the result. In this case it will be (4 + 1) x (1 + 1) = 5 x 2 = 10 (the power of 2 is 4 and the power of 3 is 1) Therefore, there will be 10 factors including 1 and 48.
Excluding, these two numbers, you will have (10 – 2) = 8 factors. And the factors are: 2, 3, 4, 6, 8, 12, 16 and 24.

The problem of factoring large numbers is so difficult that it is in fact computationaly intractable with current technology and is the basis for many modern codes.

Your "simple technique" is only simple for small numbers, where it is trivial.

[John Cuthber]

I'm not sure, but I think it might be a more efficient way of finding the number of factors of a number than the obvious (find them all then count them).
However, finding the number of factors of a number is not something I have ever had occasion to do. (As opposed to finding what the factors actually are which is relatively useful)
For example for a large number that is of the form 2^n X 3^m where n and m are large, it's quicker to find n and m than to try to divide the large number by all integers less than it's square root (though there must be better algorithms than that).

It seems to work best for number with a few prime factors raised to large powers- and those are rare.