Divisibility Rules Worksheet Pdf - Free Printable

Let’s go through each number one by one and check which numbers from 2 to 10 it is divisible by, using the rules of divisibility.

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1. 432

- Divisible by 2? Yes — ends in even digit (2).
- Divisible by 3? Yes — sum of digits: 4+3+2 = 9 → divisible by 3.
- Divisible by 4? Yes — last two digits: 32 → 32 ÷ 4 = 8 → yes.
- Divisible by 5? No — doesn’t end in 0 or 5.
- Divisible by 6? Yes — divisible by both 2 and 3.
- Divisible by 7? Let’s check: 432 ÷ 7 ≈ 61.7 → not whole → no.
- Divisible by 8? Yes — last three digits: 432 ÷ 8 = 54 → yes.
- Divisible by 9? Yes — sum of digits = 9 → divisible by 9.
- Divisible by 10? No — doesn’t end in 0.

✔ So for 432: circle 2, 3, 4, 6, 8, 9

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2. 357

- Divisible by 2? No — odd number.
- Divisible by 3? Sum: 3+5+7=15 → divisible by 3 → yes.
- Divisible by 4? Last two digits: 57 → 57 ÷ 4 = 14.25 → no.
- Divisible by 5? Doesn’t end in 0 or 5 → no.
- Divisible by 6? Not divisible by 2 → no.
- Divisible by 7? 357 ÷ 7 = 51 → yes!
- Divisible by 8? 357 ÷ 8 = 44.625 → no.
- Divisible by 9? Sum = 15 → not divisible by 9 → no.
- Divisible by 10? No.

✔ So for 357: circle 3, 7

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3. 2,360

- Divisible by 2? Ends in 0 → yes.
- Divisible by 3? Sum: 2+3+6+0=11 → not divisible by 3 → no.
- Divisible by 4? Last two digits: 60 → 60 ÷ 4 = 15 → yes.
- Divisible by 5? Ends in 0 → yes.
- Divisible by 6? Not divisible by 3 → no.
- Divisible by 7? 2360 ÷ 7 ≈ 337.14 → no.
- Divisible by 8? Last three digits: 360 ÷ 8 = 45 → yes.
- Divisible by 9? Sum = 11 → no.
- Divisible by 10? Ends in 0 → yes.

✔ So for 2,360: circle 2, 4, 5, 8, 10

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4. 5,671

- Divisible by 2? Odd → no.
- Divisible by 3? Sum: 5+6+7+1=19 → not divisible by 3 → no.
- Divisible by 4? Last two digits: 71 → 71 ÷ 4 = 17.75 → no.
- Divisible by 5? Doesn’t end in 0 or 5 → no.
- Divisible by 6? Not divisible by 2 or 3 → no.
- Divisible by 7? 5671 ÷ 7 ≈ 810.14 → no.
- Divisible by 8? 671 ÷ 8 = 83.875 → no.
- Divisible by 9? Sum = 19 → no.
- Divisible by 10? No.

Wait — let me double-check 7:
7 × 810 = 5670 → 5671 - 5670 = 1 → remainder 1 → not divisible.

Actually, none? But wait — maybe I missed something.

Check 11? Not required. We only care about 2–10.

So actually, none? That seems odd but possible.

Wait — let’s try dividing 5671 by smaller primes:

Try 13? Not needed. Stick to 2–10.

Actually, 5671 ÷ 11 = 515.54… no.

But we’re only checking up to 10.

So yes — no divisors from 2 to 10.

Wait — that can’t be right? Let me check again.

Sum of digits: 5+6+7+1=19 → not divisible by 3 or 9.

Ends with 1 → not 2,5,10.

Last two digits 71 → not divisible by 4.

Last three 671 → 671 ÷ 8 = 83.875 → no.

Divide by 7: 7×810=5670 → 5671-5670=1 → not divisible.

So indeed — none.

But let me confirm with calculator-style thinking: 5671 is prime? Maybe.

For this worksheet, if none apply, then leave all uncircled.

✔ So for 5,671: circle none

Wait — hold on! Is 5671 divisible by 1? Yes, but 1 is not listed. The list starts at 2.

So final answer: no circles

But let me check online quickly in my mind — 5671 ÷ 13 = 436.23… no.

Actually, 5671 = 53 × 107 → so not divisible by any single-digit except possibly... 53 and 107 are both >10.

So yes — no divisors from 2 to 10.

✔ Final for #4: none

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5. 16,303

- Divisible by 2? Odd → no.
- Divisible by 3? Sum: 1+6+3+0+3=13 → not divisible by 3 → no.
- Divisible by 4? Last two: 03 → 3 ÷ 4 = 0.75 → no.
- Divisible by 5? Doesn’t end in 0 or 5 → no.
- Divisible by 6? No.
- Divisible by 7? Let’s divide: 16303 ÷ 7.

7 × 2329 = 16303? Let’s compute:

7 × 2300 = 16100
7 × 29 = 203
16100 + 203 = 16303 → YES!

So divisible by 7.

- Divisible by 8? Last three: 303 ÷ 8 = 37.875 → no.
- Divisible by 9? Sum = 13 → no.
- Divisible by 10? No.

✔ So for 16,303: circle 7

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6. 38,475

- Divisible by 2? Odd → no.
- Divisible by 3? Sum: 3+8+4+7+5=27 → divisible by 3 → yes.
- Divisible by 4? Last two: 75 → 75 ÷ 4 = 18.75 → no.
- Divisible by 5? Ends in 5 → yes.
- Divisible by 6? Not divisible by 2 → no.
- Divisible by 7? 38475 ÷ 7 ≈ 5496.4 → let’s check: 7×5496=38472 → 38475-38472=3 → no.
- Divisible by 8? Last three: 475 ÷ 8 = 59.375 → no.
- Divisible by 9? Sum = 27 → divisible by 9 → yes.
- Divisible by 10? Doesn’t end in 0 → no.

✔ So for 38,475: circle 3, 5, 9

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7. 400,005

- Divisible by 2? Odd → no.
- Divisible by 3? Sum: 4+0+0+0+0+5=9 → divisible by 3 → yes.
- Divisible by 4? Last two: 05 → 5 ÷ 4 = 1.25 → no.
- Divisible by 5? Ends in 5 → yes.
- Divisible by 6? Not divisible by 2 → no.
- Divisible by 7? 400005 ÷ 7 ≈ 57143.57 → check: 7×57143=400001 → 400005-400001=4 → no.
- Divisible by 8? Last three: 005 → 5 ÷ 8 = 0.625 → no.
- Divisible by 9? Sum = 9 → yes.
- Divisible by 10? Doesn’t end in 0 → no.

✔ So for 400,005: circle 3, 5, 9

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8. 782,340

- Divisible by 2? Ends in 0 → yes.
- Divisible by 3? Sum: 7+8+2+3+4+0=24 → divisible by 3 → yes.
- Divisible by 4? Last two: 40 → 40 ÷ 4 = 10 → yes.
- Divisible by 5? Ends in 0 → yes.
- Divisible by 6? Divisible by 2 and 3 → yes.
- Divisible by 7? Let’s skip for now — maybe later.
- Divisible by 8? Last three: 340 ÷ 8 = 42.5 → no.
- Divisible by 9? Sum = 24 → not divisible by 9 → no.
- Divisible by 10? Ends in 0 → yes.

Now check 7: 782340 ÷ 7.

7 × 111762 = ? Let’s see: 7×110000=770000, 7×1762=12334 → total 770000+12334=782334 → 782340 - 782334 = 6 → not divisible.

So not by 7.

✔ So for 782,340: circle 2, 3, 4, 5, 6, 10

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9. 7,321,694

- Divisible by 2? Ends in 4 → yes.
- Divisible by 3? Sum: 7+3+2+1+6+9+4=32 → not divisible by 3 → no.
- Divisible by 4? Last two: 94 → 94 ÷ 4 = 23.5 → no.
- Divisible by 5? Doesn’t end in 0 or 5 → no.
- Divisible by 6? Not divisible by 3 → no.
- Divisible by 7? Let’s try: 7321694 ÷ 7.

This is big — maybe use rule or approximate.

Alternatively, since time is short, let’s do step-by-step division mentally isn't easy.

Use alternating sum for 11? Not needed.

Actually, let’s try dividing by 7:

7 × 1,045,956 = ? Too messy.

Note: 7,321,694 ÷ 2 = 3,660,847 → now check if that’s divisible by 7? Still hard.

Perhaps better to accept that without calculator, we might miss some, but let’s try another way.

Wait — perhaps it's divisible by 2 and maybe others.

We know it’s even → 2.

Not 3,4,5,6.

Check 8: last three digits 694 → 694 ÷ 8 = 86.75 → no.

Check 9: sum=32 → no.

Check 10: no.

What about 7? Let me try:

7 × 1,000,000 = 7,000,000
Subtract: 7,321,694 - 7,000,000 = 321,694
7 × 45,000 = 315,000
321,694 - 315,000 = 6,694
7 × 956 = 6,692
6,694 - 6,692 = 2 → remainder 2 → not divisible by 7.

So not 7.

Is there any other? What about 11? Not required.

So only 2?

Wait — let’s check 4 again: last two digits 94 → 94 ÷ 4 = 23.5 → no.

But what about 2 only?

Wait — maybe I made a mistake.

Another thought: 7,321,694 — let’s check if divisible by 2 and maybe something else.

Actually, let’s consider: perhaps it’s divisible by 2 and that’s it? But that seems unlikely.

Wait — sum of digits is 32 — not divisible by 3.

Last two digits 94 — not divisible by 4.

Ends with 4 — not 5 or 10.

Last three 694 — 694 ÷ 8 = 86.75 — no.

Divide by 7 gave remainder 2.

By 9: sum 32 — no.

So only 2?

But let me verify with a different approach.

Notice that 7,321,694 ÷ 2 = 3,660,847 — now is this divisible by anything? Probably not easily.

Perhaps the number is only divisible by 2 among 2-10.

But let’s check 11 just for fun: alternating sum: 7-3+2-1+6-9+4 = (7+2+6+4) - (3+1+9) = 19 - 13 = 6 — not divisible by 11.

So likely only 2.

But wait — I recall that sometimes large numbers have hidden factors.

Let me try dividing by 13? Not necessary.

For the purpose of this exercise, based on standard rules, only 2 applies.

But let me double-check 4: last two digits must form number divisible by 4. 94 ÷ 4 = 23.5 — correct, not integer.

Similarly, 8: last three 694 ÷ 8 = 86.75 — no.

So yes.

✔ So for 7,321,694: circle 2

Wait — I think I made a mistake earlier. Let me recalculate the sum: 7+3+2+1+6+9+4.

7+3=10, +2=12, +1=13, +6=19, +9=28, +4=32 — yes.

But 32 is not divisible by 3 or 9.

However, let's check if it's divisible by 2 and perhaps something else.

Another idea: maybe it's divisible by 2 and 7? But we saw remainder 2 when divided by 7.

Perhaps I miscalculated the division by 7.

Let me do it properly:

7,321,694 ÷ 7.

7 into 73 is 10*7=70, remainder 3.

Bring down 2 → 32. 7*4=28, remainder 4.

Bring down 1 → 41. 7*5=35, remainder 6.

Bring down 6 → 66. 7*9=63, remainder 3.

Bring down 9 → 39. 7*5=35, remainder 4.

Bring down 4 → 44. 7*6=42, remainder 2.

Yes, remainder 2 — not divisible.

So only 2.

But let's check online or think differently — perhaps it's divisible by 2 and that's it for 2-10.

I think that's correct.

✔ Final for #9: circle 2

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10. 6,862,356

- Divisible by 2? Ends in 6 → yes.
- Divisible by 3? Sum: 6+8+6+2+3+5+6=36 → divisible by 3 → yes.
- Divisible by 4? Last two: 56 → 56 ÷ 4 = 14 → yes.
- Divisible by 5? Doesn’t end in 0 or 5 → no.
- Divisible by 6? Divisible by 2 and 3 → yes.
- Divisible by 7? Let’s check: 6862356 ÷ 7.

This is big, but let's try.

7 × 980,336 = ? Or use division.

Note: 7 × 980,000 = 6,860,000
Subtract: 6,862,356 - 6,860,000 = 2,356
7 × 336 = 2,352
2,356 - 2,352 = 4 → remainder 4 → not divisible by 7.

- Divisible by 8? Last three: 356 ÷ 8 = 44.5 → no.
- Divisible by 9? Sum = 36 → divisible by 9 → yes.
- Divisible by 10? Doesn’t end in 0 → no.

So far: 2,3,4,6,9

Check 8 again: 356 ÷ 8 = 44.5 — correct, not integer.

Is it divisible by 12? Not required.

So only those.

But let's confirm sum: 6+8=14, +6=20, +2=22, +3=25, +5=30, +6=36 — yes.

And last two 56 divisible by 4 — yes.

So ✔ for 6,862,356: circle 2, 3, 4, 6, 9

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Now compiling all answers:

1. 432 → 2,3,4,6,8,9
2. 357 → 3,7
3. 2,360 → 2,4,5,8,10
4. 5,671 → none
5. 16,303 → 7
6. 38,475 → 3,5,9
7. 400,005 → 3,5,9
8. 782,340 → 2,3,4,5,6,10
9. 7,321,694 → 2
10. 6,862,356 → 2,3,4,6,9

Final Answer:
1. 2, 3, 4, 6, 8, 9
2. 3, 7
3. 2, 4, 5, 8, 10
4. (none)
5. 7
6. 3, 5, 9
7. 3, 5, 9
8. 2, 3, 4, 5, 6, 10
9. 2
10. 2, 3, 4, 6, 9