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?
The factors of the 3 answers must be
even numbers. The only odd factors will be 1.
Plans to Solve/Investigate the
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
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
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