Composite numbers greater than 2 are all the whole numbers greater than 2 that are not prime.
Definition of Composite Numbers
A composite number is a positive integer that has at least one divisor other than 1 and itself [1.1]. In other words, a composite number is a non-prime number greater than 1. All composite numbers can be expressed as a product of two or more prime numbers through a process called prime factorization. For example, the number 12 is a composite number because it has factors 1, 2, 3, 4, 6, and 12. It can be written as the product of primes, 12=2×2×312 equals 2 cross 2 cross 3
12=2×2×3
. The number 1 is considered neither prime nor composite. The number 2 is the only even prime number, as its only factors are 1 and 2. All other even numbers greater than 2 are composite because they are divisible by 2 [1.2].
Prime vs. Composite Numbers
The key difference between prime and composite numbers lies in their number of factors.
- A prime number is a natural number greater than 1 that has exactly two distinct positive divisors: 1 and itself. Examples are 2, 3, 5, 7, 11, and 13.
- A composite number is a natural number greater than 1 that has more than two positive divisors. Examples are 4, 6, 8, 9, 10, and 12.
Every integer greater than 1 is either a prime number or a composite number. The Fundamental Theorem of Arithmetic states that every integer greater than 1 can be uniquely expressed as a product of prime numbers, regardless of the order of the factors [1.1].
Examples of Composite Numbers Greater Than 2
The composite numbers greater than 2 are all even numbers except for 2, and odd numbers that are not prime. Here is a list of the first few composite numbers:
-
4 (2×22 cross 2
2×2
)
-
6 (2×32 cross 3
2×3
)
-
8 (2×2×22 cross 2 cross 2
2×2×2
)
-
9 (3×33 cross 3
3×3
)
-
10 (2×52 cross 5
2×5
)
-
12 (2×2×32 cross 2 cross 3
2×2×3
)
-
14 (2×72 cross 7
2×7
)
-
15 (3×53 cross 5
3×5
)
-
16 (2×2×2×22 cross 2 cross 2 cross 2
2×2×2×2
)
Any even number greater than 2 is composite because it is divisible by 2. Odd composite numbers include 9, 15, 21, 25, 27, and 33. All of these numbers are products of other smaller numbers (e.g., 9=3×39 equals 3 cross 3
9=3×3
, 15=3×515 equals 3 cross 5
15=3×5
) [1.2].
Identifying Composite Numbers
To determine if a number is composite, you can check if it is divisible by any prime number up to the square root of that number. For example, to check if 91 is a composite number, you only need to test for divisibility by prime numbers up to 91the square root of 91 end-root
91√
, which is approximately 9.5. The primes to check are 2, 3, 5, and 7. Since 91 is divisible by 7 (91=7×1391 equals 7 cross 13
91=7×13
), it is a composite number [1.1]. This method is much more efficient than testing every number up to the number itself.
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