# Actually Applicable Application Problems and Brainteasers/The Nines Trick for Digit Sums

## Overview

[edit | edit source]To be honest, this is a bit of a brainteaser or party trick. You can Actually Apply it as a step any time you need to know digit sums, such as for divisibility rules.

## General Method

[edit | edit source]When adding up a number's digits, you can eliminate the digit 9. For example, starting with the number 295, 2+9+5=16 and 1+6=7, and 2+5=7.

You can also eliminate pairs or groups of digits that add up to 9, like the 4 and 5 in 425 or the 1, 2, and 6 in 1,726.

### A note about why this works

[edit | edit source]The reason this trick works has to do with equivalency classes mod 9, an alternative number universe which is based on remainders when dividing by 9 in which adding 9 is the same as adding 0.

## Problems

[edit | edit source]### Divisibility

[edit | edit source]Is 438 a multiple of 3?

Is 362 divisible by 9?

### Make Your Own Problem

[edit | edit source]Find any number and add up its digits, using the nines trick to make it faster. This can be a good way to pass time in a car because you usually have a steady supply of numbers such as other cars' licence plates.