Chapter 15
STANDARD LONG DIVISION MADE EASY
In the previous chapter we saw how to divide by large numbers using factors. This principle is central to
all long division, including standard long division commonly taught in schools.
Division by factors worked well for division by numbers such as 36 (6 × 6), 27 (3 × 9), and any other
number that can be easily reduced to factors. But what about division by numbers such as 29, 31 or 37,
that can’t be reduced to factors? These numbers are called prime numbers; the only factors of a prime
number are 1 and the number itself.
Let me explain how our method works in these cases.
If we want to divide a number like 12,345 by 29, this is how we do it. We can’t use our long division
by factors method because 29 is a prime number. It can’t be broken up into factors, so we use standard
long division.
We set the problem out like this:
We then proceed as we did for short division. We try to divide 29 into 1, which is the first digit of
12,345, and of course we can’t do it. So we join the next digit and divide 29 into 12. We find that 12 is
also too small — it is less than the number we are dividing by — so we join the next digit to get 123.
Now we divide 29 into 123. This is where we have a problem; most people don’t know the 29 times
table, so how can they know how many times 29 will divide into 123?
The method is easy. This is how everyone does long division but they don’t always explain it this
way. Firstly, we round off the number we are dividing by. We would round off 29 to 30. We divide by
30 as we go, to estimate the answer, and then we calculate for 29.
How do we divide by 30? Thirty is 10 times 3, so we divide by 10 and by 3 to estimate each digit of
the answer. So, we divide 123 by 30 to get our estimate for the first digit of the answer. We divide 123
by 10 and then by 3. To roughly divide 123 by 10 we can simply drop the final digit of the number, so
we drop the 3 from 123 to get 12. Now divide 12 by 3 to get an answer of 4. Write 4 above the 3 of 123.
Our work looks like this:
Now we multiply 4 times 29 to find what the remainder will be. Four times 29 is 116. (An easy way
to multiply 29 by 4 is to multiply 30 by 4 and then subtract 4.)
4 × (30 − 1) = 120 − 4 = 116
Write 116 below 123 and subtract to find the remainder.
123 − 116 = 7
We bring down the next digit of the number we are dividing and write an X beneath to remind us the
digit has been used.