Divisibility Rules: 11-16, and Some

A continuation of divisibility rules.  For rules for the digits 0 through 10, refer to the post titled Divisibility Rules: 0-10.

11 – Alternatively add and subtract the digits of an integer, from one end to the other.  If the resultant is a multiple of 11 then the whole is a multiple of 11.  For example, take the value 365167484.  We add and subtract its digits alternatively: .  The sum is a multiple of 11 (Zero is a multiple of eleven!).  Because the resultant is a multiple of eleven, so too is the whole.  The process is reiterative.

12 – The number twelve can be factored into 3 and 4.  A number is divisible by 12 only if it is divisible by both 3 and 4, simultaneously.  Refer to those rules.

13 – The method for thirteen is not unlike the first method for seven.  So similar in fact, I will actually copy and paste it here and perform only minor edits. Compare:

“Suppose the integer s is in the form , where m and n are both integers as well, but restrict n to a single digit.  This is saying, take n to be the ones digit and take m to be everything to the left of the ones digit as its own integer.  For example, , therefore and .  The process is relatively simple.  Multiply n by nine and subtract it from m: .  In this example we take 1697 and we subtract 72 for a difference of 1625.  Now, 16978 is divisible by 13 only if 1625 is divisible by 13.  That is to say, more generally, is divisibe by 13 if is divisible by 13.  The process is reiterative.”

13 – An alternative method from the one mentioned above is such that: .  Everything else is otherwise the same.

14 – Fourteen can be factored into 2 and 7.  Thus, a number is divisible by 14 only if it is divisible by both 2 and 7. It must be divisible by 7 and be even.

15 – Fifteen can be factored into 3 and 5. Thus, a number is divisible by 15 only if it is divisible by both 3 and 5.  That is, divisible by 3 while ending in a 0 or a 5.

16 – The divisibility rules for 16 follow the same pattern as for 4 and 8.  There are two approaches.  A number must be divisible by 2 four distinct times. Or, the four least significant digits must be divisible by 16 by themselves.

2^k – To generalize the rule of 2, 4, 8, 16 and the like – If you are testing if a number, s, is divisible by n, where , for some positive integer k (such is the case with 2, 4, 8, 16, etc.), then the two tests are as follows: The number s must be divisible by 2 in total k distinct times; or the least significant k digits of s must be divisible by n.

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