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Question

Dear Mitch,

You recently answered questions about the divisibility rules for the numbers nine and the number three, and you mentioned  that there were tricks just as easy to tell if a really long number is divisible by any of the numbers from 1-10, plus a lot of bigger numbers.  My daughter and I tremendously appreciated the ones you posted, and we were wondering if you would post a couple more when you get a chance.

Thank you.

Your fans,

Mr. Fash and my daughter, Kim Fash

Answer

Dear Mr. Fash and Kim Fash,

Sure!

It would be my pleasure.

First, one of the easiest ones there is, which I'm sure you sort of know (whether you realize it or not) is the divisibility test for 5:

Any number on the planet, no mattter how big or small, if it ends with either a five or a zero, then it is evenly divisible by five (when I say 'evenly', I mean that the number five goes into the bigger number a certain number of times, and that number of times could be odd or even), AND THERE WILL BE NO REMAINDER.

All you have to do  is think of that ancient tune that little girls with pigtails often chant to themselves when jumping rope, and more often than not you will hear: "five, ten, fifteen, twenty, twenty-five, thirty, thirty-five, forty, etc...")

But an even easier test is the one for TEN.  ANY number on the planet that ends with a zero is divisible by 10.  For example, the number 333, 444, 354, 678, 540 ends with a zero, and therefore can be divided by 10, and you will end up with a nice big group of tens with no remainder.  Even the number zero, which ends with a zero, is divisible by 10 (zero times), and that equation leaves no remainder!

Here's one more for today, and I'll save the rest for another day this week.  The number 6.  To check to see if a number is evenly divisible by 6, there is a very easy test, though this test is the only one mentioned today that is two steps.  First, check to make sure the big, long number is even (in other words, check to see that it ends wih either a 0, a 2, a 4, a 6, or an 8), which, by the way, is the divisibility test for 2!), and then, if the big number is even, you perform the test for three on it (adding all the digits together and seeing if that SUM is divisible by three).  If you think about it, it makes perfect sense, as two and three are both factors of six, and if some long number is evenly divisible by some other number, then the long number is also always divisible by the factors of that smaller number.

For six, you must get a yes in both categories: the big number you are checking must be divisible by both 2 AND 3 -- then it's a YES for 6.  If either category gives you a NO, then it is a NO for 6.

I hope you found this interesting, and I hope this helped!

Mathematically yours,

Mitch