Maths - Prime Factors

Methods and tricks to solve questions related to Prime factors.

1). Counting the Number of Factors -- If you factor a number into its prime power factors, then the total number of factors is found by adding one to all the exponents and multiplying those results together.

Example: The total number of factors of 108 are --
108 = (2^2) * (3^3) thus we add 1 to the exponents and multiply the results together.

(2+1)*(3+1) = 3*4 = 12.

Hence total number of factors of 108 are 12.

Verifying the above

The factors of 108 are 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, and 108 , which are total 12 in number.


2). Factoring Factorials - First write the number you are factoring as a product of two or more numbers. For example, suppose we want to factor the number 30.

We know that 30 = 5 x 6,

so we write for the first step of our factor tree:

30
/ \
5 6

Next we factor the factors, if possible. In other words, for each number in the product from the first step, we try to write it down as a product of even smaller numbers.

For instance, in our example, we would try to factor both 5 and 6. As we noted above, 5 is prime, so we can't factor it further.

We are done with that branch of the factor tree.

However, 6 is not prime. 6 = 2 x 3, so we can extend our factor tree as follows:

30
/ \
5 6

Now further

6
/ \
2 3

Now, we continue this process until all of the branches of the tree end in prime numbers.

In our example, we are done after two steps, since 5, 2, and 3 are all prime numbers.

The factors of the number are the numbers at the end of the different branches on the factor tree. To figure out what power each prime factor is raised to, count the number of times the prime factor appears in the factor tree.

In our example, each of the factors appears only once, so the prime factorization of 30 is: 30 = 2 x 3 x 5.