Lesson 4: Common Factors and Common Multiples

Property

• A factor is a whole number that divides exactly into another number. If $a \times b = c$, then $a$ and $b$ are factors of $c$.
• A multiple of a number is the result of multiplying that number by a counting number (1, 2, 3, ...).

Examples

Property

The greatest common factor (GCF) of two numbers is the largest number that is a factor of both numbers.

Examples

Property

The smallest number that is a multiple of two numbers is called the least common multiple (LCM).
To find the least common multiple (LCM) of two numbers by listing multiples:
Step 1. List the first several multiples of each number.
Step 2. Look for multiples common to both lists. If there are no common multiples in the lists, write out additional multiples for each number.
Step 3. Look for the smallest number that is common to both lists.
Step 4. This number is the LCM.

Examples

• For 10 and 14: Multiples of 10 are $10, 20, 30, 40, ...$ and multiples of 14 are $14, 28, 42, ...$. The first common number is 70, so the $\operatorname{LCM}(10, 14) = 70$.
• For 9 and 15: Multiples of 9 are $9, 18, 27, 36, 45, ...$ and multiples of 15 are $15, 30, 45, ...$. The $\operatorname{LCM}(9, 15) = 45$.
• For 7 and 14: Multiples of 7 are $7, 14, 21, ...$ and multiples of 14 are $14, 28, ...$. The first one in common is 14, so the $\operatorname{LCM}(7, 14) = 14$.

Explanation

The LCM is the smallest positive number that is a multiple of two or more numbers. It's the first “meeting point” if you imagine listing out the multiples of each number in a sequence.