In the two previous posts, 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 11 were presented. In this post, the divisibility rules for 12 to 15 are introduced.
12: There are two divisibility rules for 12:
a) If a number is divisible by 3 and 4 then it is divisible by 12.
b) [this rule may not be very useful; it is written here only FYI] For a number, subtract the last digit from twice the rest; if the result is divisible by 12 then that number is divisible by 12.
Example: For 132 is divisible by 12 because:
(13 * 2) − 2 = 24, and 24 is divisible by 12.
13: There are two divisibility rules for 13:
a) For a number, form the alternating sum of blocks of three from right to left, if the result is divisible by 13 then that number is divisible by 13.
Example 1: 1600625 is divisible by 13 because:
Alternating sum of blocks of three from right to left:
1,600,625
625 - 600 + 1 = 26, and 26 is divisible by 13.
Example 2: 26727319 is divisible by 13 because:
347,455,147
147 - 455 + 347 = 39 (which is divisible by 13).
b) For a number, add 4 times the last digit to the rest; if the result is divisible by 13 then that number is divisible by 13.
In the following, two examples are presented for this rule, where in the second example, this rule is applied in two rounds.
Example 1: 234 is divisible by 13 because:
23 + (4 * 4) = 39 (which is divisible by 13).
Example 2: 897 is divisible by 13 because:
89 + (4 * 7) = 117
and 117 is divisible by 13 because:
11 + (4 * 7) = 39 (which is divisible by 13).
14: There are two divisibility rules for 14:
a) If a number is divisible by 2 and 7 then it is divisible by 14.
b) For a number, add the last two digits to twice the rest; if the result is divisible by 14 then that number is divisible by 14.
Example 1: 924 is divisible by 14 because:
(9 * 2) + 24 = 42 (which is divisible by 14: 3 * 14 = 42).
Example 2: 15526 is divisible by 14 because:
26 + (155 * 2) = 336
and 336 is divisible by 14 because:
36 + (3 * 2) = 42 (which is divisible by 14).
15: If a number is divisible by 3 and 5 then it is divisible by 15.