Divisibility Rules Worksheets - Math Monks - Free Printable

To solve this divisibility test, we need to check each number against the rules for 2, 3, 5, 7, 8, and 10. Here are the quick rules we will use:

* Divisible by 2: The number is even (ends in 0, 2, 4, 6, or 8).
* Divisible by 3: Add all the digits together. If the sum is divisible by 3, the number is too.
* Divisible by 5: The number ends in 0 or 5.
* Divisible by 7: Double the last digit and subtract it from the rest of the number. If the result is divisible by 7 (including 0), the original number is too.
* Divisible by 8: Look at the last three digits. If that 3-digit number is divisible by 8, the whole number is.
* Divisible by 10: The number ends in 0.

Let's go through each number step-by-step.

1. Number: 204
* 2: Ends in 4 (even). Yes.
* 3: $2 + 0 + 4 = 6$. 6 is divisible by 3. Yes.
* 5: Does not end in 0 or 5. No.
* 7: Last digit is 4. Double it ($4 \times 2 = 8$). Subtract from the rest ($20 - 8 = 12$). 12 is not divisible by 7. No.
* 8: Last three digits are 204. $204 \div 8 = 25.5$. Not a whole number. No.
* 10: Does not end in 0. No.

2. Number: 240
* 2: Ends in 0 (even). Yes.
* 3: $2 + 4 + 0 = 6$. Divisible by 3. Yes.
* 5: Ends in 0. Yes.
* 7: Last digit is 0. Double it ($0 \times 2 = 0$). Subtract from rest ($24 - 0 = 24$). 24 is not divisible by 7. No.
* 8: Last three digits are 240. $240 \div 8 = 30$. Yes.
* 10: Ends in 0. Yes.

3. Number: 1,435
* 2: Ends in 5 (odd). No.
* 3: $1 + 4 + 3 + 5 = 13$. 13 is not divisible by 3. No.
* 5: Ends in 5. Yes.
* 7: Last digit is 5. Double it ($5 \times 2 = 10$). Subtract from rest ($143 - 10 = 133$). Let's check 133. Last digit 3, double is 6. $13 - 6 = 7$. 7 is divisible by 7. So, Yes.
* 8: Last three digits are 435. 435 is odd, so it can't be divided by 8 evenly. No.
* 10: Does not end in 0. No.

4. Number: 24,724
* 2: Ends in 4 (even). Yes.
* 3: $2 + 4 + 7 + 2 + 4 = 19$. 19 is not divisible by 3. No.
* 5: Does not end in 0 or 5. No.
* 7: Last digit 4, double is 8. $2472 - 8 = 2464$.
* Check 2464: Last digit 4, double is 8. $246 - 8 = 238$.
* Check 238: Last digit 8, double is 16. $23 - 16 = 7$.
* 7 is divisible by 7. So, Yes.
* 8: Last three digits are 724. $724 \div 8 = 90.5$. No.
* 10: Does not end in 0. No.

5. Number: 65,702
* 2: Ends in 2 (even). Yes.
* 3: $6 + 5 + 7 + 0 + 2 = 20$. 20 is not divisible by 3. No.
* 5: Does not end in 0 or 5. No.
* 7: Last digit 2, double is 4. $6570 - 4 = 6566$.
* Check 6566: Last digit 6, double is 12. $656 - 12 = 644$.
* Check 644: Last digit 4, double is 8. $64 - 8 = 56$.
* 56 is divisible by 7 ($7 \times 8 = 56$). So, Yes.
* 8: Last three digits are 702. $702 \div 8 = 87.75$. No.
* 10: Does not end in 0. No.

6. Number: 106
* 2: Ends in 6 (even). Yes.
* 3: $1 + 0 + 6 = 7$. Not divisible by 3. No.
* 5: Does not end in 0 or 5. No.
* 7: Last digit 6, double is 12. $10 - 12 = -2$. Not divisible by 7. No.
* 8: Last three digits are 106. $106 \div 8 = 13.25$. No.
* 10: Does not end in 0. No.

7. Number: 36
* 2: Ends in 6 (even). Yes.
* 3: $3 + 6 = 9$. Divisible by 3. Yes.
* 5: Does not end in 0 or 5. No.
* 7: Last digit 6, double is 12. $3 - 12 = -9$. Not divisible by 7. No.
* 8: $36 \div 8 = 4.5$. No.
* 10: Does not end in 0. No.

8. Number: 1,008
* 2: Ends in 8 (even). Yes.
* 3: $1 + 0 + 0 + 8 = 9$. Divisible by 3. Yes.
* 5: Does not end in 0 or 5. No.
* 7: Last digit 8, double is 16. $100 - 16 = 84$.
* Check 84: Last digit 4, double is 8. $8 - 8 = 0$.
* 0 is divisible by 7. So, Yes.
* 8: Last three digits are 008 (which is just 8). $8 \div 8 = 1$. Yes.
* 10: Does not end in 0. No.

Final Answer:

| Number | 2 | 3 | 5 | 7 | 8 | 10 |
| :--- | :---: | :---: | :---: | :---: | :---: | :---: |
| 204 | ✓ | ✓ | | | | |
| 240 | ✓ | ✓ | ✓ | | ✓ | ✓ |
| 1,435 | | | ✓ | ✓ | | |
| 24,724 | ✓ | | | ✓ | | |
| 65,702 | ✓ | | | ✓ | | |
| 106 | ✓ | | | | | |
| 36 | ✓ | ✓ | | | | |
| 1,008 | ✓ | ✓ | | ✓ | ✓ | |