Prime Factorization

Prime Factorization Tips and Tricks

Last Updated : 15 Oct, 2024

Let us first go through the standard Prime Factorization by 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 Division Method.

Prime Factorization Example By Division Method

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

Prime Factorization using Division Method

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

Prime Factorization by Division Method

How can we make the process faster?

The idea is reduce the number faster using divisibility rules.

1. If the given number has 0s as 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 sum of digits is divisible by 3. For example 122211 is divisible by 3 because sum of digits is 9 (divisible by 3).

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

4. 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 sum of alternating digits is (1 + 1) - 2 = 0. 0 is divisible by 11

5. 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 that is why the standard RSA algorithm for encryption uses this idea.

Prime Factorization

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