145 not a prime, is a composite number. 145 = 5 × 29; Explanations below.

How to factor a number into prime factors (prime factorization)

Let's learn by an example, take the number 220 and build its prime factorization.

0) We need the first prime numbers list, let's say from 2 up to 100: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97. Prime numbers are the basic blocks of the composite numbers prime factorization process.

1) Start by dividing 220 by the first prime number, 2:
220 ÷ 2 = 110; remainder = 0 => 220 is divisible by 2 (2 | 220) => 2 is a prime factor of 220. => 220 = 2 × 110.

2) Divide the result of the previous operation, 110, by 2, again:
110 ÷ 2 = 55; remainder = 0 => 110 is divisible by 2 (2 | 110) => 2 is once again a prime factor of 220 => 220 = 2 × 2 × 55.

3) Divide the result of the previous operation, 55, by 2, again:
55 ÷ 2 = 27 + 1; remainder = 1 => 55 is not divisible by 2.

4) Divide the result of the step 2 operation, 55, by the next prime number, 3:
55 ÷ 3 = 18 + 1; remainder = 1 => 55 is not divisible by 3.

5) Divide the result of the step 2 operation, 55, by the next prime number, 5:
55 ÷ 5 = 11; remainder = 0 => 55 is divisible by 5 (5 | 55) => 5 is yet another prime factor of 220 => 220 = 2 × 2 × 5 × 11.

6) Notice that 11 is also a prime number, so we've got already all the prime factors of 220.

7) Conclusion, 220 prime factorization: 220 = 2 × 2 × 5 × 11.
This product of prime factors can be written in a condensed form, by the use of exponents: 220 = 22 × 5 × 11.

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