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 
logical!"

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 
itself.

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
  http://mathforum.org/dr.math/