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

Overview

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

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

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

Divisibility

Is 438 a multiple of 3?

Is 362 divisible by 9?

Make Your Own Problem

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.