I have run across some interesting properties of the quotient obtained from dividing 2 integers. I will list the properties in which I am interested at the conclusion of this post. The question is: Where can I find proofs of these properties.

Here are the properties:

1. An integer is divisible by 3 if the sum of its digits is divisible by 3.

2. An integer is divisible by 4 if the last 2 digits of the number form a number which is divisible by 4.

3. An integer is divisible by 8 if the last 3 digits of the number form a number which is divisible by 8.

4. An integer is divisible by 9 if the sum of its digits is divisible by 9.

Thanks for any input.

... doug