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Title
Even Factors

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

The factors of the 3 answers must be even numbers.  The only odd factors will be 1.

Plans to Solve/Investigate the Problem

Explore the factors of numbers smaller than 10 first.  Then randomly choose a larger number to factor.  Look for a pattern.

Investigation/Exploration of the Problem

The factors of 4 are 1, 2, 4, even factors except for 1, so 4 meets the criteria.  The factors of 8 are 1, 2, 4, 8, even factors except for 1, so 8 meets the criteria.  I then began to list the factors of 200, but they included 25, so 200 did not meet the criteria.  Then I tried 2.  The factors of 2 are 1 and 2, meeting the criteria.  I noticed the pattern:  2, 4, 8, so then I tried 16, 32, and 64.  Each of these numbers had even factors with only 1 as an odd factor.  If you put all of these answers in order, you notice that each number is twice its previous number.

Extensions of the Problem

You could continue to prove the pattern by going on to 128, 256, etc.

You could also see if there is a pattern for numbers with only odd factors.  The factors of 15 are 1, 3, 5, 15.  The factors of 25 are 1, 5, 25.  The factors of 19 are 1, 19; the factors of 99 are 1, 99, 3, 33, 9, 11.  The only pattern immediately noticeable is that of course this would only apply to odd numbers.

Choose another even number to look for a pattern.    The factors of 6 which are even except for 1, but the pattern does not hold for 12 since the factors of 12 include 2 odd factors.  The factors of 20 include more odd factors.  It appears that the pattern holds true only for 2, 4, 8, etc.

This discovery needs to tie into divisibility rules, either after the rules have been discussed or after.

Author & Contact
Vicki Hughes
vhughes@rockdale.k12.ga.us

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