Math is a beautiful thing. I was showing my third grade and sixth grade sons the nifty thing about the multiplication facts for the number nine. If you write them down, starting at 9 times 1 and work your way up the table there is a curious thing that occurs. The one’s place of the product decreases by one as the 10’s place increases by the same. 9, 18, 27, 36… My wife noticed that the sum of the digits was always 9. Working together we tested our little theory that maybe all numbers whose digits sum to a number divisible by 9 are in fact also divisible by 9. With a suggestion from our sixth grader, we calculated the sum of 123456789’s digits (45). Nice. Totally divisible by 9. 123456789/9=13717421. Fantastic. This worked with 3 as well. Simple since 9 is divisible by 3.
After they went to bed, I needed to look this up and see what other nifty rules of divisibility that there might be. That could give me something else to teach them later. Without getting into the Math-iness of modulo arithmetic there are indeed many different variations of determining the divisibility of a number by another number without having to perform the division out right. We already figured out the 3 and 9 version ourselves. I think everyone who can multiply or even skip count knows the rules for 2, 5, and 10 without much thought. If the one’s digit is even you can divide the number by 2. If the one’s digit is a 5 or 0 (zero) you can divide it by 5. If the one’s digit is a 0 (zero) you can divide it by 10. Easy-peasy. What came next was so very interesting that I thought writing this blog post would be fun.
1, 2, 3, 5, 9, 10 – These are easy rules to remember. We can use these rules together to figure out other numbers of divisibility with the combination of one of these. As an example, to figure out if a number is divisible by 6 we can determine if the sum of the digits are divisible by 3 and the one’s digit is even. Since, 2 x 3 = 6, the number would be divisible by 6 if it is divisible 2 AND 3 together. We have other missing divisors (4, 7, 8) before we get to 10. 4 is pretty simple. Take the last two digits if they are divisible by 4 so is the whole number. There are some really complicated rules for calculating the divisibility of 7 that uses recursive operations. A little complicated for my grade-schoolers and quite frankly might be easier to just grab your calculator. 8 is nice because you only have to evaluate the divisibility of the last three digits.
For some in-depth information check out the site Cut the Knot. There is plenty more the math gobbledygook there to make your proofs if you need them.