Divisibility Rules
Divisibility Rules are derived from Arithmetics and knowing by heart the multiplication table of the first 10 numbers and some basic additions is just what you need.
Divisibility is the division of two numbers without remainder.
The question is if a number is divisible (without remainder) by 1,2,3,4,5,6,7,8 or 9.
Here a modified table from Wikipedia which explain the divisibility rules. To check if a number is divisible by 3, check the rule under the column "divisor" and look for number "3".
Divisor 
Rule 
Examples 
1 
Automatic. 
Any integer is divisible by 1. 
2 
The last digit is even (0, 2, 4, 6, or 8). 
3576: 6 is even. 
3 
The sum of the digits is divisible by 3. 
405 => 4+0+5=9 clearly divisible by 3.
16,499,205,854,376 => 1+6+4+9+9+2+0+5+8+5+4+3+7+6 sums to 69 => 6 + 9 = 15 => 1 + 5 = 6, which is clearly divisible by 3. 
4 
The last two digits divisible by 4. 
40832: 32 is divisible by 4. 
5 
The last digit is 0 or 5. 
495: the last digit is 5. 
6 
It is divisible by 2 and by 3. 
1,458: 1 + 4 + 5 + 8 = 18, so it is divisible by 3 and the last digit is even, hence the number is divisible 6. 

7 
 Divisibility rules for 7 aren't easy to remember.
 Take the last digit in a number.
 Double and subtract the last digit in your number from the rest of the digits.
 Repeat the process for larger numbers.

357 (Double the 7 to get 14. Subtract 14 from 35 to get 21 which is divisible by 7 and we can now say that 357 is divisible by 7. 
8 
If the hundreds digit is even, examine the number formed by the last two digits and see if it is divisible by 8 
624: 24, 24 is divisible by 8 (3x8 = 24) 
If the hundreds digit is odd, examine the number obtained by the last two digits plus 4. 
352: 52 + 4 = 56. 
Add the last digit to twice the rest. 
56: (5 × 2) + 6 = 16. 
Examine the last three digits 
34152: Examine divisibility of just 152: 19 x 8 
9 
The sum of the digits is divisible by 9. 
2,880: 2 + 8 + 8 + 0 = 18 =1 + 8 = 9.

10 
The last digit is 0. 
130: the last digit is 0. 
There are other divisibility rules for higher numbers. Wikipedia offers number from 10 to 20 as well.
From divisibility rules back to fractions as measure
