I can’t give you all the ways to test for divisibility. Here are some common ones to test for divisibility by:
2: The last digit of the number is 0 or divisible by 2.
3: The sum of all the digits that make up the number is divisible by 3
4: The number formed by the last 2 digits of a number is divisible by 4
5: The last digit of the number is either 0 or 5.
8: The number formed by the last three digits of the number is divisible by 8
9: The sum of all the digits of the number is divisible by 9
11: The sum of the digits which have a place value that is an even power of ten is equal to the sum of the digits with a place value that is an odd power of 10
The rules to decide the divisibity of a number by 2 is that the given number should be even. It is easy to see that even numbers are those integers which has one of the the even digits in the unit place of the integer. The even digits are 0, 2,4,6,8. Example : 225 is not divisible by 2 as 225 has 5 in its unit place. 990 is divisible by 2 as the even digit 0 is in unit place of the number 990.
An integer number is divisible by 3 , if the sum of the digits in an integer number is divisible by 3. Example 126 is divisible by 3 as 1+2+6 = 9 is divisible by 3. The number 1081 is not divisible by 3 as the digits of 1081 sums to 1+0+8+1 = 10 which is not divisible by 3.
The utility of these rules are come to help in calculations of numbers with many digits, where it saves our time in deciding the divisibity rather actually dividing the number.