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are called composite. (Note that 1 is a special case according to these definitions: it is neither prime
nor composite!) The first few composite numbers are

4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, ...
How do we know that these numbers are composite? Because each has at least one factor, other than
1 and itself. For example, 15 has factors 3 and 5, that is, 15 = 3 · 5.
Every composite whole number has a unique prime factorization, which is its expression as the
product of its prime factors, listed in order of increasing size. For example:
4= 2 · 2 = 2 2
6= 2 · 3

8= 2 · 2 · 2 = 2 3
9= 3 · 3 = 3 2
10 = 2 · 5
2
12 = 2 · 2 · 3 = 2 · 3
14 = 2 · 7
15 = 3 · 5

16 = 2 · 2 · 2 · 2 = 2 4
(3.1)
To find prime factorizations, we use repeated division with prime divisors. Start by testing the
number for divisibility by the smallest prime, 2 (a number is divisible by 2 if its final digit is even:
0, 2, 4, 6 or 8). If it is divisible by 2, we divide by 2 as many times as possible, until we arrive at a
quotient which is no longer divisible by 2. We then repeat the procedure, starting with the last quotient
obtained, and using the next larger prime, 3 (a number is divisible by 3 if the sum of its digits is divisible
by 3 – did you know that?). Repeating again (if necessary) with the next larger prime, we eventually
arrive at a quotient which is itself a prime number. At that point, we are almost finished. We collect
all the primes that were used as divisors (if the same prime has been used more than once, it should be
collected as many times), together with the final (prime) quotient. The product of all these numbers is
the prime factorization.
Example 94. Find the prime factorization of 300.
Solution. 300, being even, is divisible by 2, so we start by dividing by 2. The steps are as follows:

300 ÷ 2= 150;
150 ÷ 2= 75; 75 is not divisible by 2; move on to 3
75 ÷ 3= 25; 25 is not divisible by 3; move on to 5
25 ÷ 5 = 5; 5 is a prime number; stop.

The primes that were used as divisors were 2, 2, 3, 5. Note that 2 was used twice, so it is listed twice.
The final prime quotient is 5. The prime factorization is the product of all those prime divisors and the
final prime quotient:
2
2
300 = 2 · 2 · 3 · 5 · 5 or 2 · 3 · 5 .
We can easily check our work by multiplying the prime factors and verifying that the product obtained
is the original number.

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