3.3 Prime factorization of natural numbers  (Page 2/2)

Some composite numbers are 4, 6, 8, 9, 10, 12, and 15.

Sample set b

Determine which whole numbers are prime and which are composite.

Practice set b

Determine which of the following whole numbers are prime and which are composite.

The fundamental principle of arithmetic

Prime numbers are very useful in the study of mathematics. We will see how they are used in subsequent sections. We now state the Fundamental Principle of Arithmetic.

Fundamental principle of arithmetic

Prime factorization

The technique of prime factorization is illustrated in the following three examples.

1. $\text{10}=5\cdot 2$ . Both 2 and 5 are primes. Therefore, $2\cdot 5$ is the prime factorization of 10.
2. 11. The number 11 is a prime number. Prime factorization applies only to composite numbers. Thus, 11 has no prime factorization.
3. $\text{60}=2\cdot \text{30}$ . The number 30 is not prime: $\text{30}=2\cdot \text{15}$ .

$\text{60}=2\cdot 2\cdot \text{15}$

The number 15 is not prime: $\text{15}=3\cdot 5$

$\text{60}=2\cdot 2\cdot 3\cdot 5$

We'll use exponents.

$\text{60}=2^2\cdot 3\cdot 5$

The numbers 2, 3, and 5 are each prime. Therefore, $2^2\cdot 3\cdot 5$ is the prime factorization of 60.

The prime factorization of a natural number

The following method provides a way of finding the prime factorization of a natural number.

The method of finding the prime factorization of a natural number

1. Divide the number repeatedly by the smallest prime number that will divide into it a whole number of times (without a remainder).
2. When the prime number used in step 1 no longer divides into the given number without a remainder, repeat the division process with the next largest prime that divides the given number.
3. Continue this process until the quotient is smaller than the divisor.
4. The prime factorization of the given number is the product of all these prime divisors. If the number has no prime divisors, it is a prime number.

We may be able to use some of the tests for divisibility we studied in [link] to help find the primes that divide the given number.

Sample set c

Practice set c

Find the prime factorization of each whole number.

Exercises

For the following problems, determine the missing factor(s).

For the following problems, find all the factors of each of the numbers.

For the following problems, determine which of the whole numbers are prime and which are composite.

For the following problems, find the prime factorization of each of the whole numbers.

Exercises for review