Technique

Unit digit of a product from its factors' unit digits

Arithmetic • Number System • Unit Digit

Statement

For positive integers the unit digit of a product depends only on the unit digits of the factors: replace each factor by its unit digit, multiply, and keep the unit digit of the running result at every stage. In particular a product is even exactly when at least one factor is even and is a multiple of

5

exactly when at least one factor is, so a product of odd numbers one of which is a multiple of

5

must end in

5

How to use it

1Reduce every factor to its unit digit first, then reduce the running product to its unit digit after each multiplication, so nothing longer than two digits is ever handled.
2A factor ending in $1$ can be skipped, since multiplying by it leaves the unit digit alone.
3A factor ending in $0$ , or an even factor together with one ending in $5$ , forces the product to end in $0$ .
4If every factor is odd and one of them ends in $5$ , stop: the product ends in $5$ whatever the remaining factors are.

Worked example

What is the unit digit of

17 \times 24 \times 33

Step-by-step

1
The unit digits of the three factors are $7$ , $4$ and $3$ .
Why: Only the last digit of each factor can affect the last digit of the product.
2
$7 \times 4 = 28$ , so the running unit digit is $8$
Why: Discard the tens; they cannot reach the unit place later.
3
$8 \times 3 = 24$ , so the unit digit is $4$

Answer

4

Watch out

It settles the last digit and nothing else, so it can confirm that a candidate answer is wrong but never that it is right. It also does not apply to a power with a large exponent, where the base's repeating cycle is what decides the digit rather than a single multiplication.

Also known as

unit digit of a product, last digit of a product, units digit of a product, multiply only the last digits, product that ends in 5