A divisibility rule is a quick check for whether a number divides evenly into another number, without doing the actual division. These rules save real time when factoring or checking primality by hand.
The Rules, 2 Through 12
Divisible by 2: the number is even (ends in 0, 2, 4, 6, or 8). Why it works: our number system is base 10, and 10 is divisible by 2, so only the last digit determines whether the whole number is even.
Divisible by 3: the sum of all digits is divisible by 3. For discussing 234: 2+3+4=9, divisible by 3, so 234 is divisible by 3 (234÷3=78). Why it works: 10 leaves a remainder of 1 when divided by 3 (10=3×3+1), and this pattern holds for every power of 10, which means each digit's contribution to divisibility-by-3 depends only on the digit itself, not its position — so summing all digits captures the full picture.
Divisible by 4: the last two digits, read as their own two-digit number, are divisible by 4. For 3,128: the last two digits are 28, and 28÷4=7 exactly, so 3,128 is divisible by 4. Why it works: 100 is divisible by 4, so every digit beyond the last two contributes a multiple of 100 (which is always divisible by 4) and doesn't affect the divisibility check.
Divisible by 5: the number ends in 0 or 5. Why it works: same logic as the rule for 2 — 10 is divisible by 5, so only the last digit matters.
Divisible by 6: the number is divisible by both 2 and 3 (check both rules above). Why it works: 6=2×3, and since 2 and 3 are coprime, a number divisible by both is necessarily divisible by their product.
Divisible by 7: no simple digit-sum trick exists, but a workable method: double the last digit, subtract it from the rest of the number, and check if the result is divisible by 7 (repeat if needed for large numbers). For 203: last digit 3, doubled is 6; remaining number 20; 20−6=14, which is divisible by 7 (14=7×2), so 203 is divisible by 7 (203÷7=29). This rule is less elegant than the others and, in practice, straightforward division is often just as fast for checking divisibility by 7.
Divisible by 8: the last three digits, read as their own number, are divisible by 8. For 5,432: last three digits are 432, and 432÷8=54 exactly, so 5,432 is divisible by 8. Why it works: 1,000 is divisible by 8, so digits beyond the last three contribute only multiples of 1,000, which don't affect the check.
Divisible by 9: the sum of all digits is divisible by 9 — the same mechanism as the rule for 3, just carried one step further, since 10 leaves remainder 1 when divided by 9 as well. For 4,563: 4+5+6+3=18, divisible by 9, so 4,563 is divisible by 9 (4,563÷9=507).
Divisible by 10: the number ends in 0. The simplest rule of all, since 10 is the base of our number system.
Divisible by 11: take the alternating sum of the digits (add the first, subtract the second, add the third, and so on), and check if the result is divisible by 11 (including 0). For 2,728: alternating sum from the right (or left, both work if done consistently) — using left to right: 2−7+2−8 = −11, which is divisible by 11, so 2,728 is divisible by 11 (2,728÷11=248).
Divisible by 12: the number is divisible by both 3 and 4 (check both rules above), since 12=3×4 and 3 and 4 are coprime.
Worked example combining several rules: is 504 divisible by 2, 3, 4, 6, 8, and 9? By 2: ends in 4, even — yes. By 3: digit sum 5+0+4=9, divisible by 3 — yes. By 4: last two digits 04=4, divisible by 4 — yes. By 6: divisible by both 2 and 3 — yes. By 8: last three digits 504, 504÷8=63 exactly — yes. By 9: digit sum 9, divisible by 9 — yes. 504 turns out to be divisible by all six.
Why These Shortcuts Matter Beyond Saving Time
Divisibility rules are the fastest way to narrow down candidate factors before committing to full trial division when finding a number's factors or checking primality — running the 2, 3, and 5 checks in a few seconds can rule out (or confirm) several potential factors before any actual division happens.
Common Mistakes Applying Divisibility Rules
Applying the digit-sum rule (correct for 3 and 9) to other divisors where it doesn't apply, like 4 or 8, is a frequent error — those rules depend on the last two or three digits specifically, not the full digit sum. For the 11 rule, losing track of which digits get added versus subtracted in the alternating sum is a common execution slip.
A worked example applying several rules together to quickly factor a number: find the small prime factors of 630 using divisibility rules first. By 2: even, yes, divisible by 2 (630÷2=315). By 3: digit sum 6+3+0=9, divisible by 3 (315÷3=105). By 3 again: digit sum of 105 is 1+0+5=6, divisible by 3 (105÷3=35). By 5: ends in 5, yes (35÷5=7). 7 is prime, stop. Using only quick digit-based checks (never a single long-division guess), 630 = 2 × 3² × 5 × 7 — the full prime factorization found almost entirely through pattern recognition rather than trial division.
Divisible by 25 and 100, two additional useful rules beyond 2-12. A number is divisible by 25 if its last two digits form 00, 25, 50, or 75. A number is divisible by 100 if it ends in 00. These extend the "check the last few digits" logic used for 4 and 8 to other powers of small primes.
Why the digit-sum rules for 3 and 9 feel almost like magic the first time you see them, and why they're not. The underlying reason (10 ≡ 1 mod 3, and 10 ≡ 1 mod 9) means every place-value position contributes its digit's face value, undiminished, to the divisibility check — a genuinely elegant consequence of choosing base 10 as our number system, not a coincidence or trick specific to those two numbers.
For the underlying skill these rules speed up, see How to Find the Factors of a Number and How to Check if a Number Is Prime.