Divisibility Rules: Test Without Dividing

Today's big idea

Each rule reads one feature: a last digit, the last few digits, or the digit sum.

Reading two digits works for 4 but not for 8, which needs the last three.

How it works

👀 What's going on?

Somebody asks whether 7,236 divides exactly by 9 and your hand reaches for long division. There is a test that settles it in about two seconds.

💡 Here's the trick

Look at the rules above. Each test reads one small feature of the number: its last digit, its last two or three digits, or the sum of all its digits. None of them needs the division itself.

🪄 How to do it

1.Find your divisor in the list above.
2.Read only the feature that rule asks for.
3.Apply the rule and answer yes or no.

✍️ Try it

Is 4,152 divisible by 8? Then: is 9,163 divisible by 11?

The rule for 8 reads the last three digits, and 152 divided by 8 is 19, so yes. For 11 the digits alternate: 9 minus 1 plus 6 minus 3 is 11, so yes again.

⚠️ Watch out

You might be thinking the test for 8 reads the last two digits, the way the test for 4 does. It reads the last three. The number 116 ends in 16, which divides by 4 but not by 8.

🌟 Why this matters

Spotting a factor at a glance is what makes simplifying fractions and finding an HCF quick rather than laborious.
Exam tip: Learn the digit-sum rules for 3 and 9 first, since they earn their keep most often.

Watch me solve one

Quanta solves it

“Watch how I think through one of these — then you try.”

THE PROBLEM

“Is 5,184 divisible by 8?”

1
Recall the rule for 8
Only the last three digits matter. 📋
2
Test the last three digits
184 ÷ 8 = 23
3
Decide
184 divides exactly → divisible.
4
Answer ✓
Yes — divisible by 8

Now you try one — same steps, different story 👇

See it in action

⚡ Time to play!

Divisible by 3 or 9? Add the digits. By 4? Check the last two digits. By 2, 5, 10? Check the last digit.

DIVISIBILITY · ROUND 1 OF 3

Which number is divisible by 3?

Quick check

✅ VERIFICATION

Which of these numbers is divisible by 6?

⚡ Finish the activity above first to answer.

Keep going!

Up next: Cryptarithm Puzzles

Same letter = same digit. Crack it column by column.

Concept 5 of 7 →