# What is a composite number? Definition. Examples of composite numbers and of prime numbers. All the composite numbers, up to 200

## 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 = 22
• 1st 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.
• 2nd Note: 23 = 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. 23 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 = 23
• 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 = 32
• 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 = 22 × 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.