Numbers and Mysticism

Grade Level: 7 - 12
Materials: One set of worksheets for each student
Objective: The student will be able to list all the divisors of a given integer;
determine whether a number is perfect, deficient, or abundant by applying the definition;
and determie whether two numbers are amicable by applying the definition.

Discussion: Investigating the mystical connotations attached to numbers is a natural way to generate
student interest in number theory. Since the Bible states that the earth was created in
six days, people felt that six was a mystical number. Likewise, the number twenty-eight
was held in esteem because it is the number of days in the lunar cycle. Even now, people
have associated the number 666 (the mark of the beast in Revelation) with certain
individuals. Before using the following worksheets, the student must be familiar with
the concept of a divisor of a number and with the notation below.

D_n = {all divisors of n}
D₈ = {1, 2, 4, 8}
D₂₄ = {1, 2, 3, 4, 6, 8, 12, 24}
D₂₈₄ = {1, 2, 4, 71, 142, 284}

SHEET 1

Consider D₂₈₄ = {1, 2, 4, 71, 142, 284}.
The proper divisors of 284 are all the divisors except the number itself.
Since 1 + 2 + 4 + 71 + 142 = 220, the sum of the proper divisors is 220.

What is the sum of the proper divisors of 24? __

Now consider the following definitions:

1. A number is called abundant if the sum of its proper factors exceeds the number.

Example: 24 is abundant since 1 + 2 + 3 + 4 + 6 + 8 + 12 = 36.

Find all the abundant numbers between 1 and 20 inclusive:

2. A number is called deficient if the sum of its proper factors is less than the number.

Example: 8 is deficient since 1 + 2 + 4 = 7.

Find all the deficient numbers between 1 and 20 inclusive:

3. A number is called perfect if the sum of its proper factors equals the number.

Example: 6 is perfect since 1 + 2 + 3 = 6.

Find a perfect number between 20 and 30:

Show that 496 is a perfect number:

4. Two numbers are amicable if each is the sum of the proper divisors of the other.

Example:
D₂₈₄ = {1, 2, 4, 71, 142, 284}, and
1 + 2 + 4 + 71 + + 142 = 220.
D₂₂₀ = {1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110, 220}, and
1 + 2 + 4 + 5 + 10 + 11 + 20 + 22 + 44 + 55 + 110 + 220 = 284.
So, 220 and 284 are amicable.

Show that 1184 and 1210 are amicable numbers:

SHEET 2

Much of the mysticism surrounding numbers dates back to the time of Pythagoras (500 B.C.). Special meanings were given to numbers depending on whether they were perfect, abundant, deficient, or amicable. For example, two people bearing numbers that are amicable would seal a perfect friendship between them. Persons or objects associated with deficient numbers were considered inferior, with abundant numbers above average, and with a perfect number -- well, you can imagine. Below you will find a number assigned to each letter of the alphabet. Using this assignment, you can decide whether a person or object is associated with a deficient, abundant, or perfect number.

			A -  1     H -  9     O - 47     V -  37
			B -  2     I - 10     P - 64     W -  66
			C -  4     J - 13     Q - 29     X -  87
			D -  5     K - 18     R - 85     Y -  90
			E -  6     L - 33     S - 80     Z - 100
			F -  7     M - 15     T - 81     
			G -  8     N - 17     U - 14

Example: The name WILLIAM
W + I + L + L + I + A + M = 66 + 10 + 33 + 33 + 10 + 1 + 15 = 168
D₁₆₈ = {1, 2, 3, 4, 6, 7, 8, 12, 14, 21, 24, 28, 42, 56, 84, 168}
1 + 2 + 3 + 4 + 6 + 7 + 8 + 12 + 14 + 21 + 24 + 28 + 42 + 56 + 84 = 312
The number is abundant since the sum of the proper divisors (312) is greater than the number (168).

Use the procedure in the example to analyze the following:

1. The name Michael _______________
2. The name Kathryn _
3. The dance Cha Cha _
4. Your name _
5. The name John Handley _
6. The name Pleacher _

SHEET 3

To compare two persons or objects, use the following rules:

1. Amicable numbers indicate a perfect relationship.

2. If the numbers are both abundant or both deficient, a good relationship exists.

3. If one number is abundant and the other is deficient, a poor relationship exists.

Example: SCHOOL and STUDENT

S + C + H + O + O + L = 80 + 4 + 9 + 47 + 47 + 33 = 220

S + T + U + D + E + N + T = 80 + 81 + 14 + 5 + 6 + 17 + 81 = 284

On Sheet 1, the numbers 220 and 284 were shown to be amicable --
No wonder they get along so well together!

Try the following:

1. Fire and Water _

2. Peanut Butter and Jelly _

3. The word President and the name GEORGE W. BUSH _

4. Your name and your desired profession _______________

The way the letters of the alphabet are assigned is arbitrary.
Try your hand at making up an assignment so that you and a friend will be amicable (220 and 284).

			A -        H -       O -       V -   
			B -        I -       P -       W -   
			C -        J -       Q -       X -   
			D -        K -       R -       Y -   
			E -        L -       S -       Z -  
			F -        M -       T -       
			G -        N -       U -