Number Fun

Perhaps you remember learning this trick in school:

First, take a number, add the digits together. Let’s try it with 123.

1+2+3 = 6

Now, we know that six divides evenly into three (6 ÷ 3 = 2) and the little number trick tells us that 123 is also therefore divisible by three.

On the other hand, the digits of 124 add up to seven. Seven is not evenly divisible by three, therefore 124 isn’t either.

You may also realise that this trick works for the number nine. You can see it in the multiplication table for nine:

2x9 = 18 (1+8 = 9); 3x9 = 27 (2+7 = 9); 4x9 = 36 (3+6 = 9).

The same concept is the basis for jokes like this:

Try this. It’s AMAZING! It profiles you based on numerology principles and psychological analysis to identify your favourite food from the options in the list below.

Write down your favourite number. Multiply it by three. Multiply by three again. Subtract 2.

Now take each digit of the number and add them together. Is the number larger than 9? If so add the digits of the new number. (If you have a very large favourite number you might have to repeat this cycle.)

Now use your final number to identify your favourite from the list below.

1. Ice-cream
2. Chocolate
3. Beer!!!
4. Lasagne
5. Fish & chips
6. Steak
7. Brussels sprouts
8. Apple pie
9. Tim Tams

OK, Good choice!

Basically, all these tricks work because we have ten different digits in our counting system which are using in a repetitive manner to form all possible numbers (which is why it is called the decimal or base 10 system). The largest single digit number is nine and so the trick works for any factor of 9 (e.g.: 1, 3 and 9).

Factor: If A is evenly divisible by B, then B is a factor of A.

Exactly why it should always work for factors of the largest single digit number is beyond my ken, but it does. We can confirm it by shifting to other counting systems.

If we had only eight digits in our counting system (octal or base 8) then the trick works for the number seven instead, because then seven would be the largest single digit number! In octal the number ‘10’ is one more than seven. In octal fourteen is written as '16'; the digits add up to seven and therefore fourteen is divisible by seven.

Computers often make use of a hexadecimal system which has 16 digits. This is represent by using the regular digits from 0 to 9 and supplementing them with the letters A to F. Thus ‘A’ is equivalent to our ten, ‘F’ is equivalent to our fifteen and ‘10’ represents the number sixteen! Since 'F' (fifteen) is the largest single digit number, the trick works for any factor of fifteen (e.g.: 1, 3, 5 and 15) when using the hexadecimal system.

So, let’s take the hexadecimal number ‘2B2,’ which equals the decimal number ‘690’, as an example. Remembering that ‘B’ equals ‘11’ in decimal, add the digits together. 2 + B + 2 = F (‘F’ = decimal 15)

Since F (15) is divisible by 3, 5 and of course 15, then 2B2 is also divisible by those numbers.

If you selected a trinary (base 3) counting system then the trick would work for the number two.

If you selected a base 5 counting system the trick would work with the number four.

Enjoy your sprouts!

(P.S. The amount added or subtracted after multiplying by 3 and 3 determines which item is selected from the list. Two is subtracted and 9 - 2 = 7 so you were forced to pick the seventh item in the list – sprouts!)

(P.P.S It doesn't work for zero or negative numbers.)