What are are the tests of divisibility of various numbers?
1 Answer
There are many divisibility tests. Here are a few, along with how they can be derived.

An integer is divisible by
#2# if the final digit is even. 
An integer is divisible by
#3# if the sum of its digits is divisible by 3. 
An integer is divisible by
#4# if the integer formed by the last two digits is divisible by 4. 
An integer is divisible by
#5# if the final digit is 5 or 0. 
An integer is divisible by
#6# if it is divisible by 2 and by 3. 
An integer is divisible by
#7# if subtracting twice the last digit from the integer formed by removing the last digit is a multiple of 7. 
An integer is divisible by
#8# if the integer formed by the last three digits is divisible by 8 (this can be made easier by noting that the rule is the same as for 4s if the hundreds digit is even, and the opposite otherwise) 
An integer is divisible by
#9# if the sum of the digits is divisible by 9. 
An integer is divisible by
#10# if the last digit is#0#
For these and more, take a look at the wikipedia page for divisibility rules .
Now, one may wonder about how to come up with these rules, or at least show that they actually will work. One way to do this is with a type of math called modular arithmetic .
In modular arithmetic, we pick an integer
What makes modular arithmetic very useful in determining divisibility rules is that for any integer
Let's use this to see why the divisibility rule for
But also, because
Thus:
Thus