Part 1: Factors of powers of 10
A common topic in mid-level aritmetic are tests to determine whether a number is divisible by a given small integer without needing to perform a full division; for example, that a number is divisible by nine if and only if the sum of its digits is also a multiple of nine. Proving these rules, however, can involve a bit of higher level math, such as modular arithmetic or other number-theoretic reasoning. Today, I will consider the simplest of divisibility tests; those for factors of powers of ten, which are simple due to our use of the base-ten number system.
First, divisiblity by ten is simply detemined by whether or not the ones digit is zero; divisiblity by 100; by whether the ones and tens digits are both zero; and so on. The reason is obvious.
Next, we see that divisibilty by the factors of 10, 2 and 5, also depend only on the ones digit: a number is divisible by 2 (even) if and only if the ones digit is even, and a number is divisible by five if and only if the ones digit is 0 or 5.
Those numbers which divide 100 but not 10 divide a number only if they divide the last two digits. This gives us the rules for 4, 20, 25, and 50.
For numbers dividing 1000 but not 100, we test the last three digits. This gives the tests for 8, 40, 125, 200, 250, and 500.
And so on. These, as we can see, are simple, and are obvious consequences of our base-ten system.
Monday Math 61