Problem
Divisibility tests for 2 through 9
Two numbers are given: 372 and 6525. The cells hold 2, 3, 4, 5, 6, 8 and 9. Shade a cell when the number is a multiple of it. Find every cell to shade for each number.
Reasoning · Grade 5-1 Divisibility Tests
Your answer
Watch out Multiple of 4 hints at 8, but the last three digits decide that separately.
How to solve
Strategy Make a Systematic List — There are exactly fourteen yes/no questions, so the safe move is to walk the seven divisors in order for each number and never skip one — that is a systematic list. To answer each question without dividing, I sort the seven divisors into three families that share one idea each: 2, 4, 5, 8 are decided by the last one, two or three digits (because 10, 100 and 1000 are already multiples of them), 3 and 9 are decided by the digit sum, and 6 is the only compound one, needing the 2-test AND the 3-test together. Each family is the same easy place-value fact used again, and a round of real divisions at the end checks all fourteen answers.
1STEP 1
Why looking at the last digits is enough for 2, 4, 5 and 8
For 2, 4, 5 and 8 only the last digits matter.
10 = 2 × 5, 100 = 4 × 25, 1000 = 8 × 125
This is just place value: a multi-digit number is thousands plus hundreds plus tens plus ones, and if the hundreds part is already a multiple of 4 you can set it aside and look only at what is left.
4.NBT.A.2Solve An Easier Related Problem2STEP 2
Why adding the digits works for 3 and 9
For 3 and 9 use the digit sum.
372 = (3 × 99 + 7 × 9) + (3 + 7 + 2)
Every place value is one more than a string of 9s, so the 'string of 9s' part never changes a 3- or 9-remainder and the digits are the whole story.
4.NBT.A.2Look For A PatternAdding the digits works for 3 and 9 because every place value is one more than a multiple of 9.
Why?
Ten, a hundred and a thousand each leave remainder 1 when divided by 9, so a digit in any place leaves just itself behind.
🧱Divisibility tests come from place valueBecause 10, 100 and 1000 sit in a fixed relation to a divisor, only certain digits matter — the last two decide 4, the digit sum decides 3 and 9.
Why?
The number is its place amounts added together, so the leftovers of the parts add up to the leftover of the whole.
🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.
3STEP 3
Run the seven tests on 372
372 is a multiple of 2, 3, 4 and 6.
3 + 7 + 2 = 12, 72 = 4 × 18, 372 = 8 × 46 + 4
Recognising multiples of a one-digit number is a Grade 4 skill; each test shrinks a question about a big number down to a question about a number you already know your times tables for.
4.OA.B.4Make A Systematic List4STEP 4
Run the seven tests on 6525
6525 is a multiple of 3, 5 and 9.
6 + 5 + 2 + 5 = 18 = 9 × 2, 25 = 4 × 6 + 1
Once a number fails the 2-test, every even divisor on the row (4, 6, 8) is settled at once — an odd number can never be a multiple of an even number.
4.OA.B.4Make A Systematic List5STEP 5
Check all fourteen decisions by actually dividing
Dividing directly confirms every decision.
372 = 6 × 62, 6525 = 9 × 725
Dividing a three- or four-digit number by a one-digit number and reading off the remainder is exactly the Grade 4 division skill, so a young solver can verify all fourteen decisions independently.
4.NBT.B.6Guess And Check6STEP 6
Shade the cells
Shade four cells and three cells.
Copying the yes/no list onto the picture is the last step — the thinking is already finished.
4.OA.B.4Make A Systematic ListAnswer
2, 3, 4, 6 / 3, 5, 9
3 + 7 + 2 = 12, 6 + 5 + 2 + 5 = 18
The two rows hang together internally. In row (1), 372 is shaded for 2 and for 3, so it must also be shaded for 6 — and it is; it is shaded for 4 but not for 8, which is consistent because 372 = 4 × 93 and 93 is odd, so the third factor of 2 simply is not there; it is shaded for 3 but not for 9, matching a digit sum of 12, which is a multiple of 3 but not of 9. In row (2), 6525 is odd, so every even cell (2, 4, 6, 8) must stay blank, and all four do; it is shaded for 9, and anything that is a multiple of 9 must also be a multiple of 3, which it is. Direct division confirms all fourteen cells: 372 = 2 × 186 = 3 × 124 = 4 × 93 = 6 × 62 and 6525 = 3 × 2175 = 5 × 1305 = 9 × 725.
Takeaway
Sort the tests into families — the last digits decide 2, 4, 5 and 8, the digit sum decides 3 and 9, and 6 just means 'passed 2 AND passed 3'.
- Why looking at the last digits is enough for 2, 4, 5 and 8
- Why adding the digits works for 3 and 9
- Run the seven tests on 372
- Run the seven tests on 6525
- Check all fourteen decisions by actually dividing
- Shade the cells