A divisibility rule is a shorthand way of discovering whether a number is divisible by a divisor without doing the division.
Here are some common use divisibility rules:
1: Any integer is divisible by 1.
2: If the last digit of a number is even that number is divisible by 2.
3: If the sum of the digits of a number is divisible by 3 then the number is divisible by 3.
Example 1: 785694 is divisible by 3, because:
7 + 8 + 5 + 6 + 9 + 4 = 39
and 39 is divisible by 3
Example 2: 5623855921236 is divisible by 3, because:
5 + 6 + 2 + 3 + 8 + 5 + 5 + 9 + 2 + 1 + 2 + 3 + 6 = 57
and 57 is divisible by 3, because:
5 + 7 = 12
and 12 is divisible by 3.
4: If the last two digits of a number is divisible by 4 then that number is divisible by 4.
Example: 75685232 is divisible by 4 because its last 2 digits; i.e. 32, is divisible by 4.
5: If the last digit of a number is a five or a zero then the number is divisible by 5.
6: If the sum of the digits of a number is divisible by 3 AND its last digit is even then that number is divisible by 6.
Example: 31434 is divisible by 6 because:
its last digit, 4, is an even number and
3 + 1 + 4 + 3 + 4 = 15; 15 is divisible by 3.
The above rule can also be stated as:
A number is divisible by 6 if it is divisible by 2 and by 3.
7: There are three divisibility rules for 7:
a) If for a number, the alternating sum of blocks of three from right to left is divisible by 7 then that number is divisible by 7.
Example 1: 5245317 is divisible by 7 because:
5,245,317
317 - 245 + 5 = 77
and 77 is divisible by 7
Example 2: 95225475312 is divisible by 7 because:
89,225,999,212
212 - 999 + 225 - 89 = -651
and -651 is divisible by 7: -651 / 7 = -93 (see the following rule to know how to quickly divide -651 by 7)
b) For a number, subtract 2 times of the last digit from the rest; if the result is divisible by 7 then that number is divisible by 7.
Example: 651 is divisible by 7 because:
65 - 1 * 2 = 63
and 63 is divisible by 7
c) [this rule may not be very useful; it is written here only FYI] For a number, add 5 times of the last digit to the rest; if the result is divisible by 7 then that number is divisible by 7.
Example 1: 651 is divisible by 7 because:
65 + (1 * 5) = 70
Example 2: 938 is divisible by 7 because:
93 + (8 * 5) = 133 and 133 is divisible by 7 because:
13 + (3 * 5) = 28; 28 is divisible by 7
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