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Title
Even Factors problem

Problem Statement
The number 10 has two even factors, 2 and 10; and two odd factors, 1 and 5. The number 12 has four even factors, 2, 4, 6, and 12; and two odd factors, 1 and 3. Find numbers whose factors are all even, except for the number 1. What do you notice about these numbers?

Problem setup

Find numbers whose factors are all even numbers except for the digit 1. Is there a pattern you notice?

Plans to Solve/Investigate the Problem

Through guess and check, I will develop a list of composite numbers whose factors are all even numbers.  I will begin will the smallest possible and grow from there.

Investigation/Exploration of the Problem

I list factors horizontally in numerical order from least to greatest.  I keep in mind my requirement of of no odd factors (the numbers has to be even) and also the divisibility rules.

2: 1, 2

4: 1, 2, 4

6 cannot work because of 3

8: 1, 2, 4, 8

10 cannot work because of 5

12 cannot work because of 3

14 cannot work because of 7

16: 1, 2, 4, 8, 16

18 cannot work because of 3

20 cannot work because of 5

22 cannot work because of 11

If I take out of my list all the impossible ones and continue with only those numbers that fit the 'rules', I will see a pattern:

2: 1, 2

4: 1, 2, 4

8: 1, 2, 4, 8

16: 1, 2, 4, 8, 16

32: 1, 2, 4, 8, 16, 32

64: 1, 2, 4, 8, 16, 32, 64

Each next number with only even factors except for one is double the previous number.  4 is 2 doubled; 64 is 32 doubled.  The list of factors from left to right is doubled also.  Ex: For 64, 1 * 2 = 2; 2 * 2 = 4; 4 * 2 = 8; 8 * 2 = 16; 16 * 2 = 32; and 32 * 2 = 64.  The next number in my list would be 128 since 64 * 2 = 128.

Extensions of the Problem

>Find numbers which have only one odd factor other than 1.

>Find numbers which have only odd factors.

>Find numbers which have an equal number of even and odd factors including 1.

Author & Contact
rrogers@rockdale.k12.ga.us