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
