Why Divisibility Matters
Knowing whether a number is divisible by something tells you about its factors. And factors are the key to a lot of other things: simplifying fractions, finding common multiples, prime factorization. So this stuff shows up constantly, way more than people expect.
In real terms: can 126 kids be split into equal groups of 9? Is this fraction already fully simplified? Is that big number prime? Divisibility rules help you answer all of that in your head, no long division needed.
Without divisibility rules: long division gives 1001 ÷ 7 = 143. Yes.
With the rule for 7: 1001 → 100 − 2(1) = 98 → 9 − 2(8) = −7 → divisible by 7. ✓
The rule gets you to the answer in two steps.
Rules for 2, 3, 4, 5, and 6
Divisible by 2 - If the last digit is even (0, 2, 4, 6, or 8), the number is divisible by 2. You don't even need to look at the rest of the number. This works because every multiple of 10 is already divisible by 2, so only that last digit matters.
7,019 → last digit 9 (odd) → not divisible by 2
Divisible by 3 - Add up all the digits. If the total is divisible by 3, so is the original number. If the digit sum is still a big number, add those digits up too. Keep going until it's obvious.
8,341 → 8 + 3 + 4 + 1 = 16 → 16 ÷ 3 is not whole → not divisible by 3
Divisible by 4 - Ignore everything except the last two digits. If those two digits form a number that's divisible by 4, the whole thing is. The reason: 100 is divisible by 4, so everything to the left of the last two digits is already accounted for.
5,614 → last two digits: 14 → 14 ÷ 4 = 3.5 → not divisible by 4
Divisible by 5 - Last digit is 0 or 5? It's divisible by 5. That's the whole rule. No other digit pattern works.
2,387 → last digit 7 → not divisible by 5
Divisible by 6 - Since 6 = 2 × 3, a number has to pass both tests. It has to be even, and its digit sum has to be divisible by 3. Fail either one and 6 doesn't divide it.
1,316 → even ✓ → 1+3+1+6 = 11 (not divisible by 3) ✗ → not divisible by 6
Rules for 9, 10, and 11
Divisible by 9 - Same idea as the rule for 3, just stricter. Add up the digits and see if you get a multiple of 9. Worth remembering: every multiple of 9 is also divisible by 3, but not every multiple of 3 is divisible by 9.
5,712 → 5 + 7 + 1 + 2 = 15 → 15 ÷ 9 is not whole → divisible by 3 but not 9
Divisible by 10 - Last digit is 0, end of story. It's just the rule for 2 and 5 combined: a number ending in 0 is automatically divisible by both, so it's divisible by 10.
73,455 → last digit 5 → divisible by 5 but not 10
Divisible by 11 - This one's a bit different. Take the digits, alternate between adding and subtracting them starting from the left, then check if the result is 0 or divisible by 11.
3,729 → (+3) + (−7) + (+2) + (−9) = −11 → divisible by 11 ✓
1,234 → 1 − 2 + 3 − 4 = −2 → not divisible by 11
Quick-Reference Table
Pin this somewhere. It's the fastest way to check divisibility without doing any actual division.
| Divisor | Rule | Example (YES) |
|---|---|---|
| 2 | Last digit is 0, 2, 4, 6, or 8 | 4,836 |
| 3 | Sum of all digits divisible by 3 | 5,712 (sum = 15) |
| 4 | Last two digits divisible by 4 | 3,928 (28 ÷ 4 = 7) |
| 5 | Last digit is 0 or 5 | 2,385 |
| 6 | Passes both 2-rule and 3-rule | 1,314 |
| 9 | Sum of all digits divisible by 9 | 8,019 (sum = 18) |
| 10 | Last digit is 0 | 73,450 |
| 11 | Alternating digit sum divisible by 11 (or = 0) | 4,983 (sum = 0) |
Using Rules to Simplify Fractions
This is where the rules really pay off. To simplify a fraction, you need a common factor for the top and bottom. Instead of guessing, just run through the divisibility rules systematically.
Test by 3: 2+5+2 = 9 (yes) and 3+1+5 = 9 (yes) → both divisible by 3
252 ÷ 3 = 84; 315 ÷ 3 = 105 → fraction is now 84/105
Test again by 3: 8+4 = 12 (yes) and 1+0+5 = 6 (yes)
84 ÷ 3 = 28; 105 ÷ 3 = 35 → fraction is now 28/35
Test by 7: 28 ÷ 7 = 4; 35 ÷ 7 = 5 → fraction is now 4/5
Result: 252/315 = 4/5
Start with the small primes: 2, then 3, then 5. Work your way up. Once neither the numerator nor the denominator shares a factor with the other, you're done. That's your fully simplified fraction.