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}
D
8 = {1, 2, 4, 8}
D
24 = {1, 2, 3, 4, 6, 8, 12, 24}
D
284 = {1, 2, 4, 71, 142, 284}
SHEET 1
Consider D
284 = {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:
D
284 = {1, 2, 4, 71, 142, 284}, and
1 + 2 + 4 + 71 + + 142 = 220.
D
220 = {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 - 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
D
168 = {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 -