Prime Factorization Tips and Tricks

Let us first go through the standard Prime Factorization by the Division Method. Then we will be talking about tips and tricks to make it faster.

Step 1: Divide the number by the smallest prime number (i.e. 2) until we are able to divide the given number without leaving any remainder.

Step 2: Move on to the next prime number and repeat the division until the quotient becomes 1.

Step 3: The prime factors are the divisors used in the division process.

Example 1: Find the Prime Factorization of 60 using the Division Method.

Example 2: Find the Prime Factorization of 210 using the Division Method.

Example 3: Express 56 as the product of its Prime Factors.

How can we make the process faster?

The idea is to reduce the number faster using divisibility rules.

1. If the given number has 0s at the end, the number is divisible by 10. So we remove 0s and add prime factors 2 x 5 for every removed 0.

2. If the last digit is 5, we know the number is divisible by 5.

3. A number is divisible by 3 if the sum of its digits is divisible by 3. For example, 122211 is divisible by 3 because the sum of its digits is 9 (divisible by 3).

4. A prime factor of a number cannot be more than the square root of the number. So we need to check only numbers less than or equal to the square root. For example, for 18, we do not need to check beyond 4.

5. To check divisibility by 11, if the difference between the sum of its odd-positioned digits and its even-positioned digits is 0 or a multiple of 11. For example, 121, the difference between the sum of alternating digits is (1 + 1) - 2 = 0. 0 is divisible by 11.

6. To check divisibility by 7, double the last digit and subtract it from the rest of the number, and if we get a number which is divisible by 7, then the whole number is divisible by 7.

Please remember, prime factorization is not easy for large numbers. It is not easy even for computers, which is why the standard RSA algorithm for encryption uses this idea.