# Composite number 4,624 prime factorization (decomposition), as a product of prime factors, in exponential form

## How to factor a number, how to break it down to prime factors (prime factorization) (Final answer, below)

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

0) We need the list of the first prime numbers, ordered from 2 up to, let's say, 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 building blocks of the composite numbers' prime factorization process.

1) Start by dividing 220 by the first prime number in our list, 2 (start with the smallest prime number). We could have skipped the division by 2 if our number were an odd number, ie. 221.
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) Move on, divide 55 by the next prime number, 3 (notice that the sum of digits of 55 is 5 + 5 = 10 is not divisible by 3, so 55 is not divisible by 3):
55 ÷ 3 = 18 + 1; remainder = 1 => as we already said, 55 is not divisible by 3.

5) Divide 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 the remaining number, 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 can be written in a condensed form, in the exponent notation: 220 = 22 × 5 × 11.

## Latest factored numbers

 4,624 = 24 × 172 Sep 27 05:49 UTC (GMT) 18,650 = 2 × 52 × 373 Sep 27 05:49 UTC (GMT) 1,960 = 23 × 5 × 72 Sep 27 05:49 UTC (GMT) 14,223,204,746 = 2 × 67 × 139 × 763621 Sep 27 05:49 UTC (GMT) 8,675,310 = 2 × 3 × 5 × 7 × 109 × 379 Sep 27 05:49 UTC (GMT) 1 is considered neither a prime nor a composite number Sep 27 05:49 UTC (GMT) 90,540 = 22 × 32 × 5 × 503 Sep 27 05:49 UTC (GMT) 322 = 2 × 7 × 23 Sep 27 05:49 UTC (GMT) 2,220,002 = 2 × 151 × 7351 Sep 27 05:49 UTC (GMT) 5,959 = 59 × 101 Sep 27 05:49 UTC (GMT) 36,107 is a prime number, it cannot be broken down to other prime factors Sep 27 05:48 UTC (GMT) 379 is a prime number, it cannot be broken down to other prime factors Sep 27 05:48 UTC (GMT) 47,991 = 3 × 17 × 941 Sep 27 05:48 UTC (GMT) see more... integer numbers factored, broken down to prime factors

## Tutoring: composite numbers prime factorization (decomposing, breaking numbers down to their prime factors)

The fundamental theorem of arithmetic says that every integer larger than 1 can be written as a product of one or more prime numbers in a way that is unique, except for the order of the prime factors.

1 is not considered prime, so the first prime number is 2. If 1 were admitted as a prime, number 15 for example could be prime factorized as 3 × 5 and 1 × 3 × 5; these two representations would be considered different prime factorizations, so the theorem above would have to be modified.

Positive integers that are only dividing by themselves and by number 1 are called prime numbers. If a number is prime, it can not be factored down to other prime factors, it is divisible only by 1 and itself; the number itself is called an IMPROPER FACTOR (improper divisor). Some people also consider 1 as an improper factor.

A composite number is a positive integer that has at least one positive factor (divisor) other than 1 and the number itself. A composite number is also any positive integer larger than 1 that is not a prime number.

A prime number can't be factored down to prime factors, but a number that is a composite can be, as it is shown bellow:

Example 1: 6 is divisible by 6, 3, 2 and 1, so 6 is not a prime, it's a composite number; 6 can be factored in different ways, as 1 × 6, or 1 × 2 × 3, or 2 × 3; but its prime factorization is always: 6 = 2 × 3.

Example 2: 120 can be factored in different ways, as 4 × 30 or 2 × 2 × 2 × 15 or 2 × 2 × 2 × 3 × 5; its prime factorization is always: 120 = 23 × 3 × 5; this is the condensed form of writing, with exponents, of the longer: 120 = 2 × 2 × 2 × 3 × 5.

It is important to know about numbers prime factorization in order to calculate the greatest common factor GCF of numbers (also called the greatest common divizor GCD, or highest common factor, HCF) - GCF is needed when reducing (simplifying) fractions to the lowest terms, or to calculate the least common multiple, LCM - this is needed when adding or subtracting ordinary fractions...

Example of prime numbers: 2 is divisible only by 2 and 1, so 2 is a prime number; 13 is divisible only by 13 and 1, so 13 is a prime number;

Please have a look at all the prime numbers, 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 used as basic blocks when building the prime factorizations of the composite numbers.