Divisibility Rules
Quick tests tell whether a number divides evenly without dividing.
About this topic
Divisibility rules are shortcuts that tell you whether one number divides another without carrying out the division. A number is divisible by 2 if it ends in an even digit, by 5 if it ends in 0 or 5, by 10 if it ends in 0, and by 3 or 9 if its digit sum is divisible by 3 or 9.
Other useful tests: a number is divisible by 4 if its last two digits form a number divisible by 4, by 6 if it passes both the 2 and 3 tests, and by 11 if the alternating sum of its digits is divisible by 11. These tests speed up factoring, simplifying fractions and spotting primes. They work because of place value: powers of ten leave remainder 1 when divided by 9, so a number behaves like the sum of its digits.
Try 7,254. It ends in 4, so it is even. The digit sum is 7 + 2 + 5 + 4 = 18, which is divisible by 3 and 9, so 7,254 is divisible by 2, 3, 6 and 9. For 1,331 the alternating sum is 1 − 3 + 3 − 1 = 0, so it is divisible by 11. Mental checks like this let you factor large numbers quickly.
The rules also help in checking arithmetic, a technique called casting out nines, and appear in barcode and ID-number check digits. A common error is using the 3 rule on the last digit rather than the digit sum, or forgetting that 7 has no simple rule that everyone remembers, so for 7 you usually just divide.
Key ideas
- Divisible by 3 if the digit sum is divisible by 3
- Divisible by 5 if it ends in 0 or 5
- Divisible by 9 if the digit sum is divisible by 9
Quick check
Use cases: where Divisibility Rules helps
- Homework: review the key idea behind Divisibility Rules before you start a problem set.
- Exam revision: use the quick check to test whether you can recall Divisibility Rules without notes.
- Visual learning: play with the interactive lab to see how Divisibility Rules changes when you adjust its inputs.
- Teaching a friend: explain Divisibility Rules in your own words, then compare your version with the key ideas.
Number and algebra skills are used in science, engineering, budgeting and computing.
Frequently asked questions
What is Divisibility Rules?
Quick tests tell whether a number divides evenly without dividing.
What will I learn in Divisibility Rules?
Divisible by 3 if the digit sum is divisible by 3. Divisible by 5 if it ends in 0 or 5. Divisible by 9 if the digit sum is divisible by 9.
Is Divisibility Rules part of Mathematics?
Yes. It is one of 200 Mathematics topics on LearnViz HighSchoolEdu, with 3 key ideas, a live animation and a quick check.
Does the Divisibility Rules lab work on a phone or tablet?
Yes. The animation resizes to fit your screen and supports touch, so you can drag, tap and slide with your finger.
How should I study Divisibility Rules?
Read the summary and key ideas first, play with the interactive lab to see the idea in motion, then answer the quick check. If you get a question wrong, go back to the key idea it tests.
Do I need an account, and is my progress saved?
No account is needed. Marking a topic as learned is saved in your own browser, so it stays on that device and is not sent to a server.