### Prime Numbers

A **prime number** is an integer greater than 1 whose only positive factors are 1 and itself.

The first ten prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23 and 29. You should memorize these common primes.

Note: The number 1 is not a prime number. Except for 2, all prime numbers are odd numbers. This means that the sum of two prime numbers will be even, unless one of the prime numbers is 2.

### Prime Factorization

Using your knowledge of prime numbers, you can break a number down into its prime factors.

You do this by finding factors using the divisibility rules. Start with the smallest factors.

140 = 2 × 70 = 2 × 2 × 35 = 2 × 2 × 5 × 7

Another way to see prime factorization is to use a factor tree. The tree shows the prime factors, in red. The prime factors of 72 are 2 × 2 × 2 × 3 × 3.

### The Greatest Common Factor (GCF)

The greatest common factor (GCF) is the largest factor (divisor) for two numbers. For example, the GCF of 10 and 15 is 5.

The primary use of GCF is to reduce or simplify fractions. (There are examples showing the GCF and fractions in the next section.)

How to get the GCF

### Factors and Multiples

A multiple of an integer n is the product of n and any integer. Sometimes it is easy to confuse factors and multiples.

The mnemonic “Fewer Factors, More Multiples” can help you remember.

For example, the factors of 12 are 1, 2, 3, 4, 6 and 12. The first 10 multiples of 12 are 12, 24, 36, 48, 60, 72, 84, 96, 108 and 120.

Factors, multiples, and divisibility are closely related. For example, these three sentences say the same thing.

Using variables: x, y, and n are integers, and y = nx. So:

### Least Common Multiple (LCM)

A *common multiple* of two or more integers is an integer that is a multiple of all of them. You can find the **least common multiple** by making a list of multiples for all the integers and identifying the smallest number that shows up on all the lists – this number will be the LCM.

The primary use of LCM is to add or subtract fractions. (There are examples showing the LCM and fractions in the next section.)

For example, these are multiples of 4:

4, 8, **12**, 16, 20, **24**, 28, 32, **36**, 40, …

Multiples of 6:

6, **12**, 18, **24**, 30, **36**, 42, 48, …

The common multiples shown in the lists are 12, 24 and 36.

The LCM is 12.

### GCF, LCM, and Prime Factors

Another way to see the GCF and LCM is to use a Venn diagram and the prime factors.

The GCF is the product of the shared factors.

The LCM is the product of all the factors, without repeating the shared factors.

For 120 and 132, the Venn diagram shows:

GCF 2 × 2 × 3 = 12

LCM 2 × 5 × (2 × 2 × 3) × 11 = 1320