Factors and Greatest Common Factors

Problem: The area of a rectangular garden is 24 square yards. List all possible whole-number dimensions the garden can have.

Solution: 1 yd x 24 yd,  2 yd x 12 yd,  3 yd x 8 yd,  and 4 yd x 6 yd

The whole-number dimensions of this rectangular garden are the factors of the number 24. The following statements can be made about the factors of 24:

• The factors of 24 are: 1, 2, 3, 4, 6, 8, 12 and 24 since each of these numbers divides exactly into 24.
• The pairs of factors of 24 are: 1 and 24, 2 and 12, 3 and 8, and 4 and 6.

To find the factors of a whole number follow this procedure:

1. Starting with 1, divide each counting number into the whole number.
2. If the numbers divide exactly (no remainder), then you have found a pair of factors.
3. List the counting number and the quotient of your division as a pair of factors.
4. Keep dividing until a factor repeats.
5. List all factors separated by commas.

Example 1: Find the factors of 12.

Counting #	Division	Factor Pair
1	12 ÷ 1 = 12	1 x 12
2	12 ÷ 2 = 6	2 x 6
3	12 ÷ 3 = 4	3 x 4
4	12 ÷ 4 = 3	4 x 3

Solution: The factors of 12 are 1, 2, 3, 4, 6 and 12.

(Note:  means that one or more factors has repeated so we stop dividing.)

Example 2: Find the factors of 20.

Counting #	Division	Factor Pair
1	20 ÷ 1 = 20	1 x 20
2	20 ÷ 2 = 10	2 x 10
3	20 ÷ 3 = 6 R2	--------
4	20 ÷ 4 = 5	4 x 5
5	20 ÷ 5 = 4	5 x 4

Solution: The factors of 20 are 1, 2, 4, 5, 10 and 20.

Example 3: Find the factors of 49.

Counting #	Division	Factor Pair
1	49 ÷ 1 = 49	1 x 49
2	49 ÷ 2 = 24 R1	--------
3	49 ÷ 3 = 16 R1	--------
4	49 ÷ 4 = 12 R1	--------
5	49 ÷ 5 = 9 R4	--------
6	49 ÷ 6 = 8 R1	--------
7	49 ÷ 7 = 7	7 x 7

Solution: The factors of 49 are 1, 7 and 49

Problem: Find the Greatest Common Factor (GCF) of 12 and 20. (Note: Place your mouse over the lists of factors below.)

The common factors of 12 and 20 are 1, 2 and 4.

The greatest common factor of 12 and 20 is 4.

Solution: GCF = 4

To find the Greatest Common Factor (GCF) of two or more whole numbers, follow this procedure:

1. Make a list of factors for each whole number.
2. Identify all factors that are common to all lists.
3. The Greatest Common Factor (GCF) is the largest of these common factors.

In Examples 4 through 6, place your mouse over the lists of factors to see the common factors.

Example 4: Find the GCF of 24 and 36.

The common factors of 24 and 36 are 1, 2, 3, 4, 6 and 12.

The greatest common factor of 24 and 36 is 12.

Solution: GCF = 12

Example 5: Find the GCF of 56 and 63.

The common factors of 56 and 63 are 1 and 7.

The greatest common factor of 56 and 63 is 7.

Solution: GCF = 7

Example 6: Find the GCF of 12, 30 and 54.

The common factors of 12, 30 and 54 are 1, 2, 3 and 6.

The greatest common factor of 12, 30 and 54 is 6.

Solution: GCF = 6

Summary: The factors of a whole number are those numbers that divide exactly into that whole number. The Greatest Common Factor (GCF) of a set of whole numbers is the largest factor common to all whole numbers in the set.

Exercises

Directions: Read each question below. Click once in an ANSWER BOX and type in your answer; then click ENTER. After you click ENTER, a message will appear in the RESULTS BOX to indicate whether your answer is correct or incorrect. To start over, click CLEAR.