Mark Porto and June Porto

Hackensack School District

TOPIC:

The purpose of this lesson is to help students understand the differences among deficient, abundant, and perfect numbers and to help them determine a few perfect numbers.

LEVEL:

Grades five through eight

TI-82 PROFICIENCY:

A basic understanding of TI-82 is required.

CLASS TIME:

Two class periods

TASK:

Step 1: Students should find the proper factors for numbers one through fifty using either the divisibility laws or TI-82.

Step 2: Students should determine the sum of the proper factors for each number. If the sum of the factors is less than the number itself, then this number is considered "deficient." If the sum of the factors is more than the number itself, then this number is considered "abundant." If the sum of the factors equals the number, then this number is considered a "perfect" number.

Note: The first perfect number is 6 and the second is 28. These are the only two perfect numbers between one and fifty. Perfect numbers are few and far between. The next perfect number after 28 jumps to 496! The next few steps provide a quick method to determine a perfect number without having to find the sum of the factors. Students should use the TI-82 calculator when following these steps.

Step 3: Starting with two as the first number, double the numbers up to 128. For example:

2, 4, 8, 16, 32, 64, 128

These are considered "starting numbers."

Step 4: List the factors of each "starting number." For example, the factors for 16 are 1,2,4,8,16. (Note: This process includes the starting number itself as a factor as opposed to the steps above which only required proper factors.)

Step 5: Write the sum of the factors of each starting number. For example, the sum of the factors for 16 are 1+2+4+8+16=31.

Step 6: If the sum of the factors of each starting number is prime, multiply this sum by the starting number. The product will be a perfect number. For example, the sum of 16 is 31, a prime number. Therefore, 31 x 16 = 496 (a perfect number).

Note: In this process, students will determine four perfect numbers; they are:

6, 28, 496, and 8128.

NOTES TO TEACHER:

This lesson on deficient, abundant, and perfect numbers is often optional in the fifth through eighth grade curriculum. However, learning these descriptive terms provides students with the concept that numbers can be described in many ways. For example, a number is odd or even, composite or prime, exponential, squared, cubed, etc. Students also will be able to note a pattern for perfect numbers.