## 1. Composite numbers definition

- A composite number is a natural number larger than 1 that has at least one divisor other than 1 and the number itself.

- The natural numbers larger than 1 that are only evenly dividing (= without a remainder) by themselves and 1 are called prime numbers.
- A composite number is also any natural number larger than 1 that is not a prime number.

## 2. The fundamental theorem of arithmetic

- The
**Prime Factorization**of a number: finding the prime numbers that multiply together to give that number. - The
**fundamental theorem of arithmetic**says that every natural number larger than 1 can be written as a product of one or more prime numbers in a way that is unique, up to the order of the prime factors. - So
**why is the number 1 not considered a prime number?**If 1 were considered a prime, then the prime factorization of the number 10, for example, could be either: 10 = 2 × 5 or 10 = 1 × 2 × 5. One would consider these two representations as two different prime factorizations of the same number, 10, so the statement of the fundamental theorem would no longer be valid.

## 3. Examples of composite numbers. Examples of prime numbers.

- According to the definition of the composite numbers,
**1 is not a composite number. 1 is also not considered a prime number either**, as we have read above, 2 and 3 are prime numbers since they are divisible only by 1 and themselves, so the first composite number is 4 (the composite numbers list starts with the number 4). - 2 is divisible only by 2 and 1, so 2 is a prime number.
- 3 is divisible only by 3 and 1, so 3 is a prime number.
- 4 is divisible by 4, 2 and 1, so 4 is not a prime number, it's a composite number. Its prime factorization is: 4 = 2 × 2 = 2
^{2} - 1
^{st}Note: The second part of the prime factorization of 4 is written using powers and exponents and it is called a condensed writing of the first part of the prime factorization of 4. - 2
^{nd}Note: 2^{3}= 2 × 2 × 2 = 8. The number 2 is called the base and 3 is the exponent. The exponent tells us how many times is the base multiplied by itself. 2^{3}is the power and 8 is the value of the power. We sometimes say that the number 2 was raised to the power of 3. - 5 is divisible only by 5 and 1, so 5 is a prime number.
- 6 is divisible by 6, 3, 2 and 1, so 6 is not a prime number, it's a composite number. Its prime factorization is: 6 = 2 × 3
- 7 is divisible only by 7 and 1, so 7 is a prime number.
- 8 is divisible by 8, 4, 2 and 1, so 8 is not a prime number, it's a composite number. Its prime factorization is: 8 = 2 × 2 × 2 = 2
^{3} - 9 is divisible by 9, 3, and 1, so 9 is not a prime number, it's a composite number. Its prime factorization is: 9 = 3 × 3 = 3
^{2} - 10 is divisible by 10, 5, 2 and 1, so 10 is not a prime number. The prime factorization of this number is: 10 = 2 × 5
- 11 is divisible only by 11 and 1, so 11 is a prime number.
- 12 is divisible by 12, 6, 4, 3, 2 and 1, so 12 is not a prime number. The prime factorization of this number is: 12 = 2 × 2 × 3 = 2
^{2}× 3

## 4. All the composite numbers, up to 200:

- 4, 6, 8, 9, 10, 12, 14, 15, 16, 18,
- 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39,
- 40, 42, 44, 45, 46, 48, 49, 50, 52, 54, 55, 56, 57, 58,
- 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78,
- 80, 81, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99,
- 100, 102, 104, 105, 106, 108, 110, 111, 112, 114, 115, 116, 117, 118, 119,
- 120, 121, 122, 123, 124, 125, 126, 128, 129, 130, 132, 133, 134, 135, 136, 138,
- 140, 141, 142, 143, 144, 145, 146, 147, 148, 150, 152, 153, 154, 155, 156, 158, 159,
- 160, 161, 162, 164, 165, 166, 168, 169, 170, 171, 172, 174, 175, 176, 177, 178,
- 180, 182, 183, 184, 185, 186, 187, 188, 189, 190, 192, 194, 195, 196, 198, 200.

- A final note on the composite numbers:
- EUCLID (300 B.C.) proved that as the set of the natural numbers is infinite, also
**the set of the prime numbers is infinite**, with no largest prime number. The same would also be true for the composite numbers. - There is no known simple formula that sets all of the composite numbers apart from the prime ones.