Part 6: Number Theory

# 6.2: Factors and multiples

Notes

##### Factors and multiples

- A
**factor**of a counting number*A*is a counting number that divides*A*evenly with no remainder (i.e.,*A*is divisible by its factors). - Suppose
*A*,*B*, and*C*are counting numbers related by the equation*A*=*B*x*C*.*B*is a factor of*A*because*A*is divisible by*B*(i.e.,*A*÷*B*=*C*).*C*is a factor of*A*because*A*is divisible by*C*(i.e.,*A*÷*C*=*B*).

**Example:***6*=*2*x*3*.*2*is a factor of*6*because*6*is divisible by*2*(i.e.,*6*÷*2*=*3*).*3*is a factor of*6*because*6*is divisible by*3*(i.e.,*6*÷*3*=*2*).

- A
**multiple**of a counting number*D*is any product of*D*and another counting number. - Suppose
*D*,*E*, and*F*are counting numbers related by the equation*D*x*E*=*F*.*F*is a multiple of*D*because*F*=*D*x*E*.*F*is a multiple of*E*because*F*=*E*x*D*.

**Example:***2*x*3*=*6*.*6*is a multiple of*2*because*6*=*2*x*3*.*6*is a multiple of*3*because*6*=*3*x*2*.

##### Find the factors of a number

**Example:**Find the factors of 126.**Method:**Repeatedly divide by counting numbers from 1, record those that divide evenly (with no remainder) together with the resulting quotient (which is also a factor), stop when a quotient would be less than the greatest factor recorded so far.- Divide by 1: factors are 1 and 126 (since 1 x 126 = 126).
- Divide by 2: factors are 2 and 63 (since 2 x 63 = 126).
- Divide by 3: factors are 3 and 42 (since 3 x 42 = 126).
- Skip 4 and 5 because they don’t divide 126 evenly.
- Divide by 6: factors are 6 and 21 (since 6 x 21 = 126).
- Divide by 7: factors are 7 and 18 (since 7 x 18 = 126).
- Skip 8 because it doesn’t divide 126 evenly.
- Divide by 9: factors are 9 and 14 (since 9 x 14 = 126).
- Skip 10, 11, 12, and 13 because they don’t divide 126 evenly.
- 14 is already in the list of factors (paired with 9).
- Stop because any number larger than 14 will have a quotient less than 9.
- The factors of 126 are 1, 2, 3, 6, 7, 9, 14, 18, 21, 42, 63, and 126.

**Example:**Find the factors of 60.- Divide by 1: factors are 1 and 60 (since 1 x 60 = 60).
- Divide by 2: factors are 2 and 30 (since 2 x 30 = 60).
- Divide by 3: factors are 3 and 20 (since 3 x 20 = 60).
- Divide by 4: factors are 4 and 15 (since 4 x 15 = 60).
- Divide by 5: factors are 5 and 12 (since 5 x 12 = 60).
- Divide by 6: factors are 6 and 10 (since 6 x 10 = 60).
- The factors of 60 are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60.

##### Find the multiples of a number

**Example:**Find the multiples of 126.**Method:**Repeatedly multiply by counting numbers from 1, record each product. There is no stopping rule because every counting number has an infinite number of multiples!- Multiply by 1: 1 x 126 = 126, so 126 is a multiple.
- Multiply by 2: 2 x 126 = 252, so 252 is a multiple.
- Multiply by 3: 3 x 126 = 378, so 378 is a multiple.
- The multiple of 126 are 126, 252, 378, …

**Example:**Find the multiples of 60.- Multiply by 1: 1 x 60 = 60, so 60 is a multiple.
- Multiply by 2: 2 x 60 = 120, so 120 is a multiple.
- Multiply by 3: 3 x 60 = 180, so 180 is a multiple.
- The multiple of 60 are 60, 120, 180, …

##### Common factors of two numbers and the GCF

- We can find common factors of two numbers.
- The largest of these is called the
*Greatest Common Factor*(GCF). - Note that the term “least common factor” is meaningless since it would always be 1.

##### Common multiples of two numbers and the LCM

- We can find common multiples of two numbers.
- The smallest of these is called the
*Least Common Multiple*(LCM). - Note that the term “greatest common multiple” is meaningless since we can continue to find greater common multiples.

##### Example: find the GCF and LCM of 126 and 60

**Slide Method:**- Find a common factor of both numbers, write the factor to the left and the resulting quotients below.
- Repeat until there are no more common factors.
- The GCF is the product of all the factors written to the left.
- The LCM is the product of the GCF and the numbers written at the bottom.

- 2 is a common factor of 126 and 60:
- write 2 to the left,
- write 126 ÷ 2 = 63 and 60 ÷ 2 = 30 below.

- 3 is a common factor of 63 and 30:
- write 3 to the left,
- write 63 ÷ 3 = 21 and 30 ÷ 3 = 10 below.

- 21 and 10 have no common factors:
- the GCF of 126 and 60 is 2 x 3 = 6,
- the LCM of 126 and 60 is 6 x 21 x 10 = 1260.

- The following figure illustrates these steps.

##### Example: find the GCF and LCM of 140 and 150

- 2 is a common factor of 140 and 150:
- write 2 to the left,
- write 140 ÷ 2 = 70 and 150 ÷ 2 = 75 below.

- 5 is a common factor of 70 and 75:
- write 5 to the left,
- write 70 ÷ 5 = 14 and 75 ÷ 5 = 15 below.

- 14 and 15 have no common factors:
- the GCF of 140 and 150 is 2 x 5 = 10,
- the LCM of 140 and 150 is 10 x 14 x 15 = 2100.

The video below works through some examples of working with factors and multiples .

Video Tips

Practice Exercises

Do the following exercises to practice working with factors and multiples.