Dear Uncle Colin,

I’ve been asked to show how to do $23241 \div 31$ in base 5 by long division - I can barely do it in base 10! Help!

- Lots Of Number Games

Hi, LONG, and thanks for your message!

There are several steps to working a long division problem, but it comes down to taking away as large a multiple of your divisor (31) as you can at each point.

The 31 times table

I would start by writing out my 31 times table in base 5:

  • $1 \times 31 = 31$
  • $2 \times 31 = 112$
  • $3 \times 31 = 143$
  • $4 \times 31 = 224$

… and that’s all we need. Now we’re going to go through the number from the front, taking off as large a ‘nice’ multiple of 31 as we can at each point.

Dividing

We start with 2. Obviously we can’t take any multiples of 31 away from that. I’d write a 0 above the 2.

What’s next? 23. We still can’t take a multiple of 31 away from that, so we write a 0 above the 3 as well.

Now we get to 232, which is bigger than 224 - so we can take away 4 lots of 31. Now you write a 4 above the 2 and figure out what you have left. Under the 232, write 224; take them away (in base 5) to get 3 and write that beneath both. Carry the remaining digits down; it should look like this:

0 0 4 —

31)2 3 2 4 1 2 2 4 —

– 3 4 1

Now the next number we have to look at is 34, which is just three more than one 31. Write a 1 above the 4, and 31 in the appropriate place below:

0 0 4 1 —

31)2 3 2 4 1 2 2 4 —

– 3 4 1 3 1 —

– 3 1

And lastly, we’ve got one 31 left over, so we write a 1 above the final 1, show there’s no remainder, and say “bingo! It’s 411.”

0 0 4 1 1 —

31)2 3 2 4 1 2 2 4 —

– 3 4 1 3 1 —

– 3 1 3 1 —

       0

Check!

We can always check this in base 10! We’ve got $2\times5^4 + 3 \times 5^3 + 2 \times 5^2 + 4 \times 5 + 1$, which is $1250 + 375 + 50 + 20 + 1$, or $1696$. Meanwhile, 31 works out to be 16.

The answer we’re looking for is $4\times 5^2 + 1 \times 5 + 1 = 106$.

And, without even bothering to wake up the Mathematical Ninja ((“I’m not asleep.”)), we can see that $1696 \div 16 = 106$, as we hoped.

I hope that helps!

- Uncle Colin