Multiplication by a multi-digit number – complete guide

In fifth grade you already know how to multiply a multi-digit number by a single digit. The next step is multiplying two multi-digit numbers — for example $1234 \times 56$ or $7234 \times 5689$ . The same trick works: split the bottom factor into separate digits, multiply the top factor by each one, and add the partial results.

What you will learn

In this topic you will work through:

The principle of long multiplication – why we split the bottom factor and shift each partial product.
Step-by-step procedure – exactly what to write at each step.
Worked examples and common mistakes – fully solved problems and the slip-ups to watch for.

A quick look at the layout

For $1234 \times 56$ you write the factors below each other and one partial product per digit of the bottom factor:

The first partial product ( $1234 \times 6 = 7404$ ) sits in the ones-and-up columns. The second one ( $1234 \times 5 = 6170$ ) is shifted one place to the left because the digit it represents is in the tens column. Adding both lines gives the answer.

Why does it work?

Splitting $56$ into its place values is the key:

1234 \times 56 = 1234 \times (50 + 6) = 1234 \times 50 + 1234 \times 6

So multiplying by $50$ is the same as multiplying by $5$ and writing the result one place to the left. That is exactly the shift you see in the column layout.

Practise

When you are ready, try the interactive exercise:

Long multiplication with multi-digit numbers – generates two factors of 2–4 digits, with empty fields for every partial product and the final sum.