Prime factorization. Prime and composite numbers
[1] The prime factorization of a number (the decomposition into prime factors): finding the prime numbers that multiply together to make that number.
Example: 12 = 2 × 2 × 3 = 22 × 3.
[2] Prime number: a natural number that is divisible (it is divided without a remainder) only by 1 and itself. A prime number has only two factors: 1 and the number itself.
Example: 2, 3, 5, 7, 11, 13, 17, 19, 23.
The only even prime number is 2. All the other prime numbers are odd numbers.
[3] Composite number: a natural number that has at least one factor other than 1 and itself. A composite number has at least three factors. A composite number is also a number that is not a prime number.
Example: 4, 6, 8, 9, 10, 12, 14, 15, 16.
The composite numbers are made of prime numbers that are multiplied together.
6,830,144 is not a prime number but a composite one.
The prime factorization (the decomposition into prime factors) of the composite number 6,830,144:
[1] The prime factorization written as a product of prime factors:
6,830,144 = 2 × 2 × 2 × 2 × 2 × 2 × 106,721
[2] The prime factorization written in canonical form = a condensed way, as a product of powers (at least some prime factors are raised to an exponent): *
6,830,144 = 26 × 106,721
The prime factorization of a number, how is it done?
Let's learn by having an example:
Take the number 220 and build its prime factorization
We need the list of the first prime numbers, ordered from 2 up to, let's say, 20:
2, 3, 5, 7, 11, 13, 17, 19.
The prime numbers are the building blocks of the composite numbers.
1. Start by dividing 220 by the smallest prime number, 2:
220 ÷ 2 = 110; remainder = 0 ⇒
220 is divisible by 2 ⇒ 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 is a prime factor of 110:
220 = 2 × 110 = 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 to the next prime number, 3. Divide 55 by 3:
55 ÷ 3 = 18 + 1; remainder = 1 ⇒
55 is not divisible by 3.
5. Move on to the next prime number, 5. Divide 55 by 5:
55 ÷ 5 = 11; remainder = 0 ⇒
55 is divisible by 5 ⇒ 5 is a prime factor of 55:
220 = 2 × 2 × 55 = 2 × 2 × 5 × 11.
6. Notice that the remaining factor, 11, is a prime number, so we've already found all the prime factors of 220.
Conclusion, the prime factorization of 220:
220 = 2 × 2 × 5 × 11.
This can be written in a condensed form, in exponential notation:
220 = 22 × 5 × 11.