Fill in the blank with the number that completes each factoring rule.

1. The number [tex]\(\qquad\)[/tex] is a factor of any number that ends with 0 or 5.
2. The number [tex]\(\qquad\)[/tex] is a factor of any even number.
3. The number [tex]\(\qquad\)[/tex] is a factor if both 2 and 3 are also factors.
4. The number [tex]\(\qquad\)[/tex] is a factor of any number whose digits add up to 9 or a multiple of 9.
5. The number [tex]\(\qquad\)[/tex] is a factor of any number whose digits add up to 3 or a multiple of 3.

List all the factors of these numbers from least to greatest.

6. 20 [tex]\(\qquad\)[/tex]
[tex]\(\qquad\)[/tex]
[tex]\(\qquad\)[/tex]
[tex]\(\qquad\)[/tex]
[tex]\(\qquad\)[/tex]
[tex]\(\qquad\)[/tex]

7. 36 [tex]\(\qquad\)[/tex]
[tex]\(\qquad\)[/tex]
[tex]\(\qquad\)[/tex]
[tex]\(\qquad\)[/tex]
[tex]\(\qquad\)[/tex]
[tex]\(\qquad\)[/tex]
[tex]\(\qquad\)[/tex]
[tex]\(\qquad\)[/tex]

Reduce these fractions to equivalent fractions in lowest terms.

8. [tex]\(\frac{15}{20}=\)[/tex]
9. [tex]\(\frac{16}{18}=\)[/tex]
10. [tex]\(\frac{4}{8}=\)[/tex]
11. [tex]\(\frac{32}{40}=\)[/tex]

Remember, to find equivalent fractions, we can either multiply or divide both the numerator and denominator by the same number. Find the missing numerator.

13. [tex]\(\overline{5}=\frac{9}{15}\)[/tex]
14. [tex]\(\frac{2}{3}=\frac{}{12}\)[/tex]
15. [tex]\(\frac{1}{2}=\overline{20}\)[/tex]
16. [tex]\(\overline{6}=\frac{20}{24}\)[/tex]
17. [tex]\(\overline{4}=\frac{6}{8}\)[/tex]
18. [tex]\(\frac{4}{7}=\frac{}{35}\)[/tex]

Write these fractions in order from least to greatest.

[tex]\(\begin{array}{lllll}\frac{3}{5} & \frac{3}{8} & \frac{3}{4} & \frac{3}{10} & \frac{3}{7}\end{array}\)[/tex]
[tex]\(\qquad\)[/tex]
[tex]\(\qquad\)[/tex]
[tex]\(\qquad\)[/tex]

Change these improper fractions to whole or mixed numbers. Be sure your answers are in the lowest terms.

23. [tex]\(\frac{5}{2}=\)[/tex]
24. [tex]\(\frac{18}{6}=\)[/tex]
25. [tex]\(\frac{21}{3}=\)[/tex]
26. [tex]\(\frac{20}{8}=\)[/tex]
27. [tex]\(\frac{10}{4}=\)[/tex]
28. [tex]\(\frac{15}{12}=\)[/tex]

Sure! Let's go through the solution step by step for each part of the question:

### Factoring Rules:

1. The number that is a factor of any number that ends with 0 or 5:
The answer is 5. Any number that ends in 0 or 5 can be divided evenly by 5.

2. The number that is a factor of any even number:
The answer is 2. All even numbers can be divided evenly by 2.

3. The number that is a factor if both 2 and 3 are also factors:
The answer is 6. If a number has 2 and 3 as factors, it can also be divided evenly by 6 (since 6 = 2 × 3).

4. The number that is a factor of any number whose digits add up to 9 or a multiple of 9:
The answer is 9.

5. The number that is a factor of any number whose digits add up to 3 or a multiple of 3:
The answer is 3.

### Listing Factors:

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

7. Factors of 36:
The factors of 36 are: 1, 2, 3, 4, 6, 9, 12, 18, and 36.

### Reducing Fractions:

8. Reducing [tex]\(\frac{15}{20}\)[/tex]:
The reduced form is [tex]\(\frac{3}{4}\)[/tex].

9. Reducing [tex]\(\frac{16}{18}\)[/tex]:
The reduced form is [tex]\(\frac{8}{9}\)[/tex].

10. Reducing [tex]\(\frac{4}{8}\)[/tex]:
The reduced form is [tex]\(\frac{1}{2}\)[/tex].

11. Reducing [tex]\(\frac{32}{40}\)[/tex]:
The reduced form is [tex]\(\frac{4}{5}\)[/tex].

### Finding Missing Numerator or Denominator:

13. Find the missing numerator in [tex]\( \frac{5}{x} = \frac{9}{15}\)[/tex]:
The missing numerator is 8.333 (approximately).

14. Find the missing denominator in [tex]\( \frac{x}{12} = \frac{2}{3}\)[/tex]:
The missing denominator is 18.

15. Find the missing numerator in [tex]\( \frac{1}{2} = \frac{x}{20}\)[/tex]:
The missing numerator is 10.

16. Find the missing numerator in [tex]\( \frac{x}{6} = \frac{20}{24}\)[/tex]:
The missing numerator is 5.

17. Find the missing numerator in [tex]\( \frac{x}{4} = \frac{6}{8}\)[/tex]:
The missing numerator is 3.

18. Find the missing denominator in [tex]\( \frac{4}{x} = \frac{7}{35}\)[/tex]:
The missing denominator is 20.

### Ordering Fractions From Least to Greatest:

- [tex]\(\frac{3}{10}, \frac{3}{8}, \frac{3}{7}, \frac{3}{5}, \frac{3}{4}\)[/tex]

### Converting Improper Fractions to Whole or Mixed Numbers:

19. [tex]\(\frac{5}{2}\)[/tex]:
- The mixed number is [tex]\(2\)[/tex] with a remainder of [tex]\(1\)[/tex], which is [tex]\(2 \frac{1}{2}\)[/tex].

24. [tex]\(\frac{18}{6}\)[/tex]:
- Converts completely to [tex]\(3\)[/tex].

25. [tex]\(\frac{21}{3}\)[/tex]:
- Converts completely to [tex]\(7\)[/tex].

26. [tex]\(\frac{20}{8}\)[/tex]:
- The mixed number is [tex]\(2\)[/tex] with a remainder of [tex]\(4\)[/tex], which is [tex]\(2 \frac{4}{8}\)[/tex] or [tex]\(2 \frac{1}{2}\)[/tex].

27. [tex]\(\frac{10}{4}\)[/tex]:
- The mixed number is [tex]\(2\)[/tex] with a remainder of [tex]\(2\)[/tex], which is [tex]\(2 \frac{2}{4}\)[/tex] or [tex]\(2 \frac{1}{2}\)[/tex].

28. [tex]\(\frac{15}{12}\)[/tex]:
- The mixed number is [tex]\(1\)[/tex] with a remainder of [tex]\(3\)[/tex], which is [tex]\(1 \frac{3}{12}\)[/tex] or [tex]\(1 \frac{1}{4}\)[/tex].

These steps explain how we arrive at the final answers for the questions provided.