1. EXAMPLE 1 Writing Factors Members of the art club are learning to do calligraphy. Their first project is to make posters to display their new lettering style. A poster will display 36 characters in order: the 26 uppercase letters of the alphabet and the digits 0 through 9. The art club members want the characters arranged in a rectangular display with the same number of characters in each row. You can use the factors of 36 to determine how many arrangements are possible. Lettering

2. EXAMPLE 1 Writing Factors In the situation above, how many ways can the art club arrange the 36 characters in a rectangular display with rows of equal length? SOLUTION STEP 1 List the factors of 36 by writing 36 as a product of two numbers in all possible ways. 1 36 2 18 3 12 6 4 9 The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, and 36.

3. EXAMPLE 1 Writing Factors Use these factors to find all the possible rectangular arrangements of rows and columns. STEP 2 1 36 2 18 3 12 6 4 9 36 1 18 2 12 3 9 4 There are nine possible rectangular arrangements. ANSWER

4. GUIDED PRACTICE for Example 1 Write all the factors of the number. 1. 20 2.29 SOLUTION The factors of 20 are 1, 2, 4, 5, 10, and 20. SOLUTION The factors of 29 are 1 and 29.

5. GUIDED PRACTICE for Example 1 Write all the factors of the number. 3.42 4.57 SOLUTION The factors of 42 are 1, 2, 3, 6, 7, 14, 21, and 42. SOLUTION The factors of 36 are 1, 3, 19 and 57.

6. GUIDED PRACTICE for Example 1 5. What If? Suppose the poster in Example 1 was to include both uppercase and lowercase letters as well as the digits 0–9. How many ways can the 62 characters be arranged in rows of equal length? ANSWER 62 characters can be arranged in 4 ways.