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Divides or not

Does 7 divide 4,571? Which of 2, 3, 4, 9, 11 divide 81,312? Answer by the rules, not by long division — up to 7, 11 and 13, with no limit on digits.

  • Number sense
  • Timed
  • Endless levels

How it plays

A number and a divisor: say whether it divides exactly. Or a number and a set of divisors: pick every one that divides it. The rules make it quick — the last digit for 2, 5 and 10, the digit sum for 3 and 9, the last two digits for 4 and 25.

Levels bring in 6, 8, 11, 12, and then 7 and 13, add a digit without end, and shorten the time. For 7, 11 and 13 the trick is 1001 = 7 × 11 × 13: split the number into groups of three from the right and add and subtract them in turn.

The working states the rule, works it on the number, and checks it by the division itself.

Keys

Space starts and skips ahead · 1–9 pick · Enter submits · Esc stops. On a phone, tap.

What it trains

This trains the divisibility rules and their use at speed. It is practice at recognising factors, not a wider claim.

Learn

How to get better at divides or not: the method, the techniques, what each new level asks of you, and the traps the wrong answers are made of.

Your approach to each question

  1. Look at the divisor first and name its rule before you read the number: “last digit”, “digit sum”, “last two”, “last three”, “alternating sum” or “groups of three”.
  2. Read only the part of the number that rule needs. For 4 you need two digits of 81,312, not five.
  3. Work the rule to a small number you can judge at a glance — 15, 12, 567 — and decide from that.
  4. When you pick several divisors, start with 2 and 3. A “no” there rules out every divisor built on it; a “yes” for 12 brings 2, 3, 4 and 6 with it.

Techniques and tricks

  • Last digits for 2, 4, 8, 5, 25, 10

    10 is 2 × 5, so only the end of a number matters for these: the last digit for 2, 5 and 10, the last two digits for 4 and 25, the last three for 8.

    e.g. 81,312: 12 = 4 × 3, so 4 divides it

  • Digit sum for 3 and 9

    Add the digits; the number divides by 3 or 9 exactly when that sum does. Add again if the sum is still large.

    e.g. 81,312: 8 + 1 + 3 + 1 + 2 = 15. That divides by 3 but not by 9.

  • Cast out nines

    While adding digits for 3 or 9, skip every 9 and every pair that makes 9. What is left decides it, and it is far less to add.

    e.g. 9,182,736: strike the 9, then 1 + 8, 2 + 7 and 3 + 6. Nothing is left, so 9 divides it.

  • Alternating sum for 11

    From the right, add and subtract the digits in turn. The number divides by 11 when the result is a multiple of 11 — and 0 and negative multiples count.

    e.g. 81,312: 2 − 1 + 3 − 1 + 8 = 11, so 11 divides it

  • 1001 for 7 and 13

    1001 = 7 × 11 × 13. Split the number into groups of three from the right — the commas already do it — and add and subtract the groups in turn. The small result keeps the remainder.

    e.g. 4,571: 571 − 4 = 567 = 7 × 81, so 7 divides it. 13 × 43 = 559 and 13 × 44 = 572, so 13 does not.

  • Halve to test 8

    Take the last three digits and halve them up to three times. If you can halve three times and stay whole, 8 divides the number.

    e.g. 8,172: 172 → 86 → 43, which is odd after two halvings, so 4 divides it but 8 does not

  • Coprime parts for 6 and 12

    Split the divisor into parts that share no factor, and test each: 6 needs 2 and 3; 12 needs 4 and 3.

    e.g. 18 passes for 2 and for 6, yet 18 ÷ 12 leaves 6. The 4 test catches it: 18 is not a multiple of 4.

As the levels climb

  • The divisors unlock in groups: 2, 5 and 10; then 3 and 9; then 4, 25 and 6; then 8, 11 and 12; finally 7 and 13. Each new group is asked about more often while it is new, so learn its rule before the level arrives.
  • A digit is added every few levels, without end. The end-digit rules do not care how long the number is, so for 2, 4, 5, 8, 10 and 25 the length changes nothing. For 3, 9 and 11, cast out nines and pair digits as you read.
  • For 7 and 13 on long numbers, the comma groups are your working: add the first, third and fifth groups from the right, subtract the second and fourth, then test the small result.
  • The clock closes to 62% of its starting time. Yes-or-no gives 3.5 seconds plus half a second per digit; the pick format adds 2.2 seconds for each divisor shown, so test the cheap rules first and let them rule out the expensive ones.

Traps to avoid

  • The near miss. Every “no” in yes-or-no is a true multiple plus a little, so it ends or sums almost right. Finish the rule; do not stop at “looks even”.
  • Two digits for 8. 116 ends in 16, which 8 divides, but 116 = 8 × 14 + 4. For 8 you always need the last three digits.
  • 2 and 6 taken as a test for 12. They share a factor of 2, so passing both proves only 6. Use 4 and 3.
  • The digit sum used for 4 or 11. The digit sum only works for 3 and 9; 4 looks at the end, 11 alternates.
  • An alternating sum of 0, or −11, read as “no”. Both are multiples of 11.

How to practise

  • Set “Format” to “Does k divide N? Yes or no” while a new group of divisors is arriving: one rule at a time is how the rule becomes automatic.
  • Switch to “Pick every divisor of N” to learn how the rules lean on each other, and raise “Divisors shown (pick format)” towards 8 once five feels easy.
  • When you miss, read the working: it is the rule worked on that very number, then the division as a check. Run the rule yourself on it before the next question.

Levels

Each question has a clock that comes from the work it asks for. There is no last level: at least one dial turns without end. Your level is judged on your last ten answers, every ten: 90% moves you up, below 80% moves you down, which holds most people near four in five right.

All of it is in the settings beside Start — how many answers a judgement looks at and how often it recalibrates, the thresholds, the time, which questions, how many options, and whether you see each verdict at once or review them together at the judgement. You can also jump to any level.

More number sense

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