## Prime numbers. Composite numbers. The prime factorization of composite numbers (decomposing, breaking down numbers into prime factors)

- The
**Prime Factorization**of a number: finding the prime numbers that multiply together to make that number. - 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.
- The number 1 is not considered prime, so
**the first prime number is 2**. - If the number 1 were considered a prime number, then the prime factorization of the number 15 could be written as: 15 = 3 × 5 OR 15 = 1 × 3 × 5 - these two representations would be considered different prime factorizations of the same number, so the theorem above would have no longer been valid.

**The natural numbers that are evenly dividing only by 1 and themselves are called prime numbers**.- Examples of prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31 and so on.
- If a number is prime, it cannot be factored down to other prime factors, it is divisible only by 1 and itself - the number itself is called in this case an IMPROPER FACTOR (or an improper divisor). Some people also consider 1 as an improper factor.

**A composite number is a natural number that has at least one factor (divisor) other than 1 and the number itself.**- A composite number is also any natural number larger than 1 that is not a prime number.
- Examples of composite numbers: 4, 6, 8, 9, 10, 12, 14, 16, 18, 20, 21, 22, 24, 25, 26 and so on.

- A prime number can't be factored down to other prime factors, but a number that is a composite can, as shown below:
- 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 written as a product of factors in different ways, as: 6 = 1 × 6, or 6 = 1 × 2 × 3 or 6 = 2 × 3. But its prime factorization, regardless the order of the factors, is always: 6 = 2 × 3.
- Example 2: 120 can be written as a product of factors in different ways, as 120 = 4 × 30 or 120 = 2 × 2 × 2 × 15 or 120 = 2 × 2 × 2 × 3 × 5. Its prime factorization is always: 120 = 2 × 2 × 2 × 3 × 5 = 2
^{3}× 3 × 5 - the last form of writing is the condensed form, with exponents, of the first form, the longer one. *** Note:**2^{3}= 2 × 2 × 2 = 8. We are saying that 2 was raised to the power of 3. In this example, 3 is the exponent and 2 is the base. The exponent indicates how many times the base is multiplied by itself. 2^{3}is the power and 8 is the value of the power.

**Why is it important to know about the prime factorization of the numbers?**- The prime factorization is useful when calculating the greatest common factor, GCF, of numbers (also called the greatest common divisor GCD, or the highest common factor, HCF).
- GCF is needed when reducing (simplifying) fractions to the lowest terms.
- The prime factorization comes in handy when calculating the least common multiple, LCM, of numbers - this is needed when adding or subtracting ordinary fractions, for example...
- And the examples could continue (numbers divisibility, calculating all the factors of a number starting from its prime factorization, and so on...).

**More examples of prime numbers:**- 181 is divisible only by 181 and 1, so 181 is a prime number.
- 2,341 is divisible only by 2,341 and 1, so 2,341 is a prime number.
- 6,991 is divisible only by 6,991 and 1, so 6,991 is a prime number.
**This is the list of all the prime numbers, from 1 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.- The prime numbers are used as basic blocks when building the prime factorization of the composite numbers. So we could say that the prime numbers really are the basic blocks of the composite numbers.