Mathematics, Algebra and Function

Grade 5- 8

### Objective

Students learn about factors, perfect numbers, abundant numbers, and defective numbers,

### Directions

Introduce to students the following information.

Perfect Numbers

Facts and Reminders

Proper Factors

The proper factors of a number are all of the factors of a number except the number itself.

**Sample A**
The factors of 14 are 1, 2, 7, and 14. The proper factors are 1, 2, and 7. (Note: The number itself, 14, is not included in the proper factors.)

**Sample B**
The factors of 9 are 1, 3, and 9. (Note: The 3 is not repeated.) The proper factors are 1 and 3.

Numbers may be designated perfect, defective, or abundant based on the sum of their proper factors.

**Perfect Numbers**
A perfect number is equal to the sum of all of its factors, except the number itself. Read the example below.

Sample

The proper factors of 6 are: 1, 2, and 3. Add 1 + 2 + 3 = 6. Therefore, 6 is a perfect number.

There are no known odd perfect numbers. Perfect numbers are very rare.

**Abundant Numbers**
Abundant numbers are those numbers where the sum of the proper factors is greater than the number itself.

**Sample**
The proper factors of 12 are 1, 2, 3, 4, and 6. Add 1 + 2 + 3 + 4 + 6. The sum, 16, is greater than 12 and 12 is, therefore, abundant. The number 12 is the first abundant number.

There are only 21 abundant numbers between 12 and 100.

**Defective (Deficient) Numbers**

Defective numbers are those numbers in which the sum of the proper factors is less than the number itself. Defective numbers are sometimes called deficient numbers.

**Sample**
The proper factors of 22 are 1, 2, and 11. Add 1 + 2 + 11. The sum, 14, is less than 22 and 22 is, therefore, defective.

Most numbers are defective because they have very few factors. All prime numbers are defective.

Distribute the activity sheets and have students use the concepts they have learned to complete the activity.

### Resources

- pencils
- activity sheets (3 per student)