In the previous post, a divisibility rule was defined as "a shorthand way of discovering whether a number is divisible by a divisor without doing the division", and the divisibility rules for 1 to 7 were presented. In this post, the divisibility rules for 8 to 11 are introduced.
8: There are four divisibility rules for 8:
a) If for a number, the third digit from right, i.e. the hundreds digit, is even then that number is divisible by 8 if the number formed by its last two digits is divisible by 8. And if the hundreds digit is odd then that number is divisible by 8 if the number formed by the last two digits plus 4 is divisible by 8.
Example 1: 5232432 is divisible by 8, because its hundreds digit, 4, is even and the number formed by its last two digits, 32, is divisible by 8.
Example 2: 985624328 is divisible by 8, because its hundreds digit, 3, is odd and the number formed by its last two digits plus four, i.e. 28 + 4 = 32, is divisible by 8.
b) For a number, if the number formed by its last three digits is divisible by 8 then that number is divisible by 8.
Example: 51248952480 is divisible by 8 because 480 (formed by the last three digits) is divisible by 8.
c) A number is divisible by 8 if the following number is divisible by 8:
(4 * [the hundreds digit]) + (2 * [the tens digit]) + [the ones digit]
Example: 987352 is divisible by 8 because:
(4 * 3) + (2 * 5) + 2 = 24 is divisible by 8
Rem: In this examle, the hundreds digit is 3, the tens digit is 5, and the ones digit is 2.
d) [this rule may not be very useful; it is written here only FYI] For a number, add the last digit to twice the rest; if the result is divisible by 8 then that number is divisible by 8.
Example: 168 is divisible by 8 because:
8 + (2 * 16) = 40
and 40 is divisible by 8.
9: If the sum of the digits of a number is divisible by 9 then the number is divisible by 9.
Example: 3375 is divisible by 9 because:
3 + 3 + 7 + 5 = 18
and 18 is divisible by 9.
10: A number is divisible by 10 if its last digit is 0.
11: There are at least four divisibility rules for 11:
a) A number is divisible by 11 if the alternating sum of its digits is divisible by 11.
Example: 937893 is divisible by 11 because:
9 - 3 + 7 - 8 + 9 - 3 = 11
and 11 is divisible by 11!
b) For a number, sbtract the last digit from the rest; if the result is divisible by 11 then that number is divisible by 11.
Example: 517 is divisible by 11 becuase:
51 - 7 = 44
and 44 is divisible by 11.
c) For a number, add the digits in blocks of two from right to left; if the result is divisible by 11 then that number is divisible by 11.
Example 1: 4125 is divisible by 11:
4125 ==> 41,25
25 + 41 = 66
and 66 is divisible by 11.
Example 2: 13354 is divisible by 11:
13354 ==> 1,33,54
54 + 33 + 1 = 88
and 88 is divisible by 11.
d) If reversing the digits of a number results in a number that is divisible by 11 then that number is also divisible by 11. In other words, if a number is divisible by 11, reversing its digits results in a number that is also divisible by 11.
Example: In the last example above, we found that 13354 is divisible by 11; so by reversing its digits, we get a number, 45331, which is divisible by 11.
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