Divisibility by nine and remainders

You may wonder why divisibility tests are important. If you wish to be quick and efficient in maths then recognizing divisibility is important when simplifying fractions, simplifying or solving algebraic equations and for pattern recognition...plus more! I think most people know that any number divisible by 2 will end in 0,2,4,6,8 (even numbers), divisible by 5 will end in 0 or 5 and divisible by 10 will end in 0. And, with a little bit of thought, you'll probably be able to work out the remainder of any number divided by 2, 5 or 10. But what about other numbers? Digital roots, once again, allow us to quickly work out, not only, if a number is divisible by 9 but also the remainder - if necessary. Let's try the number 3644. 3+6+4+4=17, 1+7=8. This number is NOT exactly divisible by 9, it would have a remainder of 8. In fact I can also tell you that 3645 is the next number divisible by 9 because 8+1=9. And that 3636 is the prior number divisible by 9 (just subtract 8). You should also note that any number divisible by 9 has a digital root of 9 - the remainder is 0 not 9 in this case. Now, for the 'true digital root' or 9's digital root. This method makes it quicker and easier to work out the remainder or divisibility of a number by 9, particularly if you have a pen and paper handy and can write down the number and cross off digits. The rule is simple - 'any digit which is nine is crossed off and replaced by zero'. You can even extend this further by saying any combination of digits that sum to 9, 18, 27 etc. can also be crossed off! I can now write a number like 1234567892 on paper and within a few seconds tell you that this number has a remainder of 2 if divided by 9. How? First cross off the 9, then 1+8, 2+7, 3+6, 4+5 and all you have left is 2. Or, if you prefer, as soon as the sum equals or exceeds 9 - subtract 9 (or add the two digits) and continue. Remember, with this method, you can never be left with 9 because it is simply crossed off and replaced with zero - automatically giving you the correct remainder!

In fact, you should never have to add ANY sums bigger than 8+8=16, because 16-9 or 1+6=7. You can simply continue with 7 added to the next digit etc. For example with 567862. 5+6=11, 1+1=2, 2+7=9 (becomes 0), 8+6=14, 1+4=5, 5+2=7. Or maybe it's easier with, 5+6+7=18 (becomes 0), 8+6+2=16, 1+6=7. You decide!