A divisibility rule peeks at a number’s digits and tells you whether a division would come out clean — remainder zero — without doing any division at all. None of it is magic: every rule falls straight out of place value.
⚡The Seven Rules, On One Card
Rules for 2, 5, and 10 hide in the LAST digit. The rule for 4 hides in the last TWO. The rules for 3 and 9 hide in the digit sum. And 6 just borrows two other rules at once.
2Last digit is even — ends in 0, 2, 4, 6, or 8.3,578 ✔ · ends in 8
3Digit sum is divisible by 3. Add up every digit and test the total.258 → 2+5+8=15 ✔
4Last TWO digits form a number divisible by 4 (because 100 already is).7,316 ✔ · 16 = 4×4
5Last digit is 0 or 5.1,295 ✔
6Passes 2 AND 3. Even, and the digit sum divides by 3.132: even ✔, 1+3+2=6 ✔
9Digit sum is divisible by 9 — the 3-rule’s stricter big sibling.4,671 → 18 ✔ · 18=9×2
10Last digit is 0. The easiest rule in mathematics.88,880 ✔
⚠️Combine rules with care! You may only stack two rules when the divisors share no common factor. 2 & 3 give you the rule for 6 — but 2 & 4 do NOT give you 8 (12 passes both 2 and 4, yet 12 ÷ 8 doesn’t come out clean). The rule for 8 needs the last THREE digits.
🔬Why the Digit-Sum Trick Works (Honestly)
The 3-rule feels like magic until you look at place value. Every 10 is a 9 plus 1. Every 100 is a 99 plus 1. So when you split a number into hundreds, tens, and ones, the 9s and 99s are already divisible by 3 — only the leftover 1-per-digit needs checking.
Same skeleton for every rule: what do 10, 100, and 1,000 leave behind when you divide them? For 2, 5, 10 they leave nothing (so only the last digit matters); for 4, only 100 leaves nothing (last two digits); for 3 and 9, each place leaves exactly 1 per digit — the digit sum.
🏃The Three-Second Check in Action
The same three-second check packs boxes, deals cards, seats banquets, and speeds up factoring and simplifying fractions — spot that 3 divides both 258 and 963 before you ever touch a pencil.
🔢 Even or odd is the 2-rule in disguise: even numbers end in 0, 2, 4, 6, 8 and split into pairs perfectly; odd numbers end in 1, 3, 5, 7, 9 and always leave a lonely 1 behind. A divisibility rule is really just a remainder-of-zero detector.
🔑Key Terms
🧼DivisibilityOne number is divisible by another when the division comes out perfectly clean — a whole-number answer with nothing left over.
⚡Divisibility RuleA fast digit-based test that tells you whether one number divides another evenly — without actually dividing.
➕Digit SumThe total from adding up every digit. The digit sum of 4,671 is 4 + 6 + 7 + 1 = 18.
👯EvenA whole number divisible by 2. Even numbers always end in 0, 2, 4, 6, or 8.
🕺OddA whole number NOT divisible by 2 — ends in 1, 3, 5, 7, or 9 and always leaves a remainder of 1.
🍰RemainderWhatever is left over after dividing as evenly as you can. A divisibility rule is a remainder-of-zero detector.
➗QuotientThe answer to a division problem — how many times the divisor fits into the number.
🔩FactorA number that divides another evenly. Since 3 divides 258 with no remainder, 3 is a factor of 258.
👟MultipleThe result of multiplying by a whole number. 36 is a multiple of 9, and 9 is a factor of 36 — same fact, two directions.
🏛️Place ValueThe value a digit carries because of where it sits — the 5 in 3,564 means 5 hundreds. The hidden engine behind every rule.
🌍Where You’ll See This in Real Life
🛒Barcodes & ID numbersThe last digit of a UPC barcode or a book’s ISBN is a check digit, chosen so the whole number passes a divisibility test. A scanner runs the test instantly — if it fails, a digit was misread and the scanner refuses to beep. Divisibility quietly guards every checkout line on Earth.
🧑🍳Splitting groups fairlyEqual teams, packed boxes, banquet tables, dealt cards — cafeteria managers and event planners run the 2-and-3 check constantly without ever calling it math.
🧾The digit-sum rule powers an ancient error-check called “casting out nines,” which merchants used to catch arithmetic mistakes — described in mathematical writing for well over a thousand years, long before calculators.
🔟The 3-and-9 trick only works because we write numbers in base ten, and 9 is one less than 10. In base eight, the very same “add the digits” test would work for 7 instead.
7️⃣There IS a rule for 7 — just fiddly: double the last digit, subtract from the rest. For 672: chop the 2, double to 4, then 67 − 4 = 63 = 7 × 9. So 672 divides by 7 (it’s 96).
📌Remember This
1The rules aren’t magic — they’re place value. Every rule is a statement about what 10, 100, and 1,000 leave behind when you divide them.
2Know where each rule looks: 2, 5, 10 → the END of the number. 4 → the last TWO digits (8 → last three). 3 and 9 → the DIGIT SUM. 6 → the rules for 2 and 3 combined.
3Only combine rules whose divisors share no common factor: 2 and 3 give you 6, but 2 and 4 do NOT give you 8.
🤔 Think about it
The rule for 8 checks the last THREE digits, not two. Using what you know about 100 and 1,000, can you explain exactly why two digits are not enough?
Suppose people wrote numbers in base twelve. Which divisor would get the “add up the digits” rule in that world — and what does that tell you about where the rule really comes from?
⭐Remember: a number’s digits are constantly confessing its secrets. Learn where each rule looks — last digit, last two, digit sum — and you’ll know whether a division comes out clean before your pencil ever touches the page.