Tests for Divisibility

Test for Divisibility:
This again is a very basic topic.
Most of u know the tests to check divisibility.
But how many of us know why those particular methods are employed ?
After the class, the logic behind the tests look so simple and I feel dumb
for not having realised this earlier.I guess most of you know the reason
as well. But I am sure there will be a fair number who don't know why.
For their sake, I am making this post.

For 3,9
Everybody knows that for 3 and 9 we find the digitsum and see if that
digit sum is divisible by 3 or 9 , whatever the case maybe.
Let us take a 2 digit number.
A two digit can be written as 10A+B
10A+B=9A+A+B
Now, 9A is always divisible by 3/9.So whether the number is divisible by 3 or 9
depends on A+B.
Any number will be of the form 10 ^ 3 A + 10 ^ 2 B + 10 C+10^0 D .....................
If we divide each co-efficient (i.e 10 ^ 0 , 10 ^ 1 , 10 ^ 2 ............) by 9, the remainder is +1.
That is why we add all the digits.

For 11,7..
Taking the previous example.
10 ^ 3 A + 10 ^ 2 B + 10 C+10^0 D
If we divide 10^0 by 11 remainder is +1.
if we divide 10 ^ 1 by 11 remainder is -1 or (10)
If we divide 10 ^ 2 by 11 remainder is + 1
If we divide 10 ^ 3 by 11 remainder is -1.
We see that for 11, it alternates between + and -.
That is why we use the difference of the sum of odd terms and sum of even terms and see if that difference
is divisible by 11.
For 7.
10 ^ 3 A + 10 ^ 2 B + 10 C+10^0 D
If we divide 10^0 by 7 remainder is +1.
if we divide 10 ^ 1 by 7 remainder is 3
If we divide 10 ^ 2 by 7 remainder is + 2
If we divide 10 ^ 3 by 7 remainder is -1 or (6)
Here we see the same pattern as in 11, but for every 3 digits.
Thats why for 7 we use difference of sum of alternate triplets.
The same holds good for 13 .

For 5,25,125.......
The first power of ten that exactly divides 5 is 1.
That is why we use the last digit of a number to see if it is
divisible by 5.
The first power of ten that exactly divides 25 is 2.
That is why we use the last 2 digits of a number to see if it is
divisible by 25
The first power of ten that exactly divides 125 is 3.
That is why we use the last 3 digits of a number to see if it is
divisible by 125

The first power of ten that exactly divides 4 is 2.
That is why we use the last 2 digits of a number to see if it is
divisible by 4
The first power of ten that exactly divides 8 is 3.
That is why we use the last 3 digits of a number to see if it is
divisible by 8