KS3 · Maths
Divisibility tests (2, 3, 4, 5, 6, 8, 9, 10)
Is 345,394,191 a multiple of 3? You could spend ages dividing. Or you could add up its digits and know for sure in about ten seconds.
Tests for 4 and 8
Reason it through
Why does the test for 4 look only at the last two digits — but the test for 8 needs the last three?
First link · your turn
Take a big number, 38,834,864. How could you split it into two parts using place value?
Number · 2, 4 and 8
How far in do you need to look?
Sort each even number into the highest group it reaches. Use the last three digits for 8, the last two for 4, and the ones digit for 2.
Still to sort
Multiple of 8 (0)
The last three digits make a multiple of 8.
Where the line is: To be here the last three digits must be a multiple of 8. Passing the test for 4 isn't enough.
Multiple of 4, but not 8 (0)
The last two digits make a multiple of 4, but the last three don't make a multiple of 8.
Where the line is: The last two digits pass for 4, but check all three before you promote it to 8.
Multiple of 2 only (0)
Even, but the last two digits aren't a multiple of 4.
Where the line is: Even is enough for 2. For 4 you need the last two digits to be a multiple of 4.
Every number here is even. The question is how far up the chain it goes.
WHAT YOU'VE LEARNED
A quick recap of today's lesson.
Is it a multiple? Read the digits and you'll know — no dividing needed.
What you need to know
- Divisible means it divides exactly with no remainder. 24 is divisible by 3 because 24 ÷ 3 = 8. 12 isn't divisible by 8 because 12 ÷ 8 = 1.5, which isn't a whole number.
- A divisibility test is a quick check on the digits that tells you whether a number is a multiple, without dividing. Dividing takes longer, but when the number is a multiple it also gives you the other factor.
- 2: the ones digit is 0, 2, 4, 6 or 8. 5: the ones digit is 0 or 5. 10: the ones digit is 0.
- 3: the digit sum is a multiple of 3. 9: the digit sum is a multiple of 9. If the digit sum is big, add its digits again.
- 6: it passes the tests for 2 AND 3. 10 works the same way with 2 and 5.
- 4: the last two digits are a multiple of 4 (because 4 is a factor of 100). 8: the last three digits are a multiple of 8 (because 8 is a factor of 1,000 but not of 100).
- The tests only check 2, 3, 4, 5, 6, 8, 9 and 10. A number that passes none of them can still have a factor greater than 10, so it isn't automatically prime.
The big picture
A number is divisible by another if it divides exactly, with no remainder. Divisibility tests answer that from the digits, without dividing, which saves time. The ones digit decides 2, 5 and 10; the digit sum decides 3 and 9; the last two digits decide 4 and the last three decide 8, because 4 is a factor of 100 and 8 is a factor of 1,000. Tests combine — a multiple of 6 passes the tests for 2 and 3 — but they only check their own divisors, so failing them all doesn't make a number prime.
Key points
Worked example
Problem
Without dividing, find which of 2, 3, 4, 5, 6, 8, 9 and 10 are factors of 7,380.
⚠ Watch out
Checking only the ones digit for 4. 58 ends in 8, which is a multiple of 4 — but 58 ÷ 4 = 14.5, so 58 isn't. The test for 4 always uses the last two digits.
Memory hook
The END for 2, 5 and 10 · the last TWO for 4 · the last THREE for 8 · ADD the digits for 3 and 9 · and 6 needs 2 AND 3.
Check yourself
Without dividing, decide which of 2, 3, 4, 5, 6, 8, 9 and 10 go into 4,536. For each one, say which digits you looked at and why.
Flashcards
(13)What does 'divisible by' mean?
Why use a divisibility test instead of dividing?
The test for 2
The test for 3
The digit sum is still big (say 135). What now?
The test for 6
The test for 9
The tests for 5 and 10
The test for 4 — and why it works
The test for 8 — and why it needs three digits
Three ways to check whether three digits make a multiple of 8
Every multiple of 4 is a multiple of 2. Is every multiple of 2 a multiple of 4?
A number fails the tests for 2, 3, 5, 6, 9 and 10. Is it prime?
Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.
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Keep me postedMore KS3 Maths topics
- Adding and subtracting positive and negative integers
- Algebraic notation and conventions
- Angle sum in a triangle
- Angles on parallel lines (corresponding, alternate, co-interior)
- Area of a circle
- Area of a trapezium
- Area of a triangle
- Area of composite rectilinear shapes
- Arithmetic sequences and finding the nth term
- Associative, commutative and distributive laws
- Bisecting an angle
- Calculating theoretical probabilities
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