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This essential trains for: Math Kangaroo 3-4, Math Kangaroo 5-6, Math Kangaroo 7-8, SAT-I, GMAT, AMC-8, AMC-10.
We will start with divisibility by 9 as the one by 3 follows from it as well.
Again, the most useful tool is to write the number in expanded form:
and now write the powers of 10 using their closest multiple of 9:
Apply the distributive property of multiplication:
Apply the commutative property of addition to re-arrange the terms:
Factor a 9 out of the last terms, to emphasize the part that is a multiple of 9:
Notice that the largest part of the number is automatically divisible by 9 (and therefore also by 3). The left over terms are the sum of the digits of the number and, if the sum of the digits is divisible by 9 (or by 3) the entire number is divisible by 9 (or by 3).
This procedure is called "casting out nines" and is applicable to other divisors as well, though in a less spectacular manner.
Notice how the rules of divisibility are highly dependent on the base of numeration that we use.