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Divisibility Rule for 12

Date: 07/10/2000 at 21:45:08
From: Laura Perry
Subject: Divisibility rules

I don't understand why the divisibility rule for 12 does not include 
2. The rule states that 12 is divisible by 3 and 4, but not 2. I got 
that question wrong on a test and I just don't understand why. After 
all, if twelve is divisible by 3 and 4, then it is also divisible by 2 
(since 2 is a factor of 4). As Spock would say, "that is just not 

Date: 07/10/2000 at 22:08:26
From: Doctor Peterson
Subject: Re: Divisibility rules

Hi, Laura.

There's some reason for including 2, as you say, but not in the rule 

The purpose of the "rule" is to let you easily decide whether a number 
is divisible by 12. The rule is not "12 is divisible by 3 and 4," but 
"a number is divisible by 12 if and only if it is divisible by both 3 
and 4." That is, if we show that it is divisible by 3 and by 4, then 
we have shown that it is divisible by 12. Remember that, if it is 
divisible by 4, then it is automatically divisible by 2 as well, so we 
don't need to test for divisibility by 2 separately; and if it is 
divisible by 2, that doesn't help us to show that it is divisible by 
12, because we still have to show it's divisible by 4 anyway. So a 
test for 2 would be redundant.

On the other hand, it is very useful to check for divisibility by 2 
before starting the test for 12. That's because it's so easy to do, 
and if it fails - if you find the number is odd - then you don't have 
to bother with the hard parts. So although testing for divisibility by 
2 doesn't help positively, it does help negatively, by screening out 
numbers that aren't worth testing for 4.

Is it logical now?

- Doctor Peterson, The Math Forum
Associated Topics:
Middle School Division

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