Numbers have habits. Learn the divisibility tests and you can tell at a glance whether 4,73,916 divides by 9 — without dividing anything.
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Each test looks at a different part of the number. None of them needs long division.
🔍 The tests worth memorising
| Divisible by | Test | Example |
|---|---|---|
| 2 | last digit is even | 3 46 ✅ |
| 3 | digit sum divides by 3 | 4 7 1 → 12 ✅ |
| 4 | last two digits divide by 4 | 5 316 ✅ |
| 5 | ends in 0 or 5 | 2 4 5 ✅ |
| 6 | passes both the 2 and the 3 test | 5 1 6 ✅ |
| 8 | last three digits divide by 8 | 9 136 ✅ |
| 9 | digit sum divides by 9 | 4 7 3 9 1 6 → 30 ❌ |
| 10 | ends in 0 | 4 5 0 ✅ |
| 11 | alternating digit sums differ by 0 or a multiple of 11 | 9 0 8 2 ✅ |
Careful with 6. A number must pass both the 2-test and the 3-test. 9 is divisible by 3 but not by 2, so it is not divisible by 6.
A factor divides the number. A multiple is in its table. HCF is the biggest shared factor; LCM is the smallest shared multiple.
🔗 Finding both by prime factors
Break both numbers into primes. 36 = 2×2×3×3, 60 = 2×2×3×5.
HCF = multiply the primes they share: 2×2×3 = 12.
LCM = take every prime at its highest power: 2×2×3×3×5 = 180.
A shortcut worth knowing: for any two numbers, HCF × LCM = the product of the numbers. Check it: 12 × 180 = 2160, and 36 × 60 = 2160. ✅ If a question gives you three of the four, this finds the fourth instantly.
Which one does the question want? "Largest tiles / greatest number that divides" → HCF. "Bells ring together again / smallest number divisible by" → LCM.
A prime has exactly two factors: 1 and itself. Everything else above 1 is composite.
1 is neither prime nor composite — it has only one factor. And 2 is the only even prime; every other even number has 2 as an extra factor. Both facts are asked directly.
The primes under 50: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47. That is 15 of them — worth learning by heart.
12 questions on divisibility, factors and multiples. 🍀
