Divisibility Rule worksheet for practicing division checks.

To solve the problem, we need to determine whether each number in the left column is divisible by the numbers at the top of each column (2, 3, 4, 5, 6, 7, 8, 9, and 10). We will use the divisibility rules for each of these numbers. Let's go through each number step by step.

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Divisibility Rules Recap:

1. Divisible by 2: The number must be even (last digit is 0, 2, 4, 6, or 8).
2. Divisible by 3: The sum of the digits of the number must be divisible by 3.
3. Divisible by 4: The last two digits of the number must form a number divisible by 4.
4. Divisible by 5: The last digit must be 0 or 5.
5. Divisible by 6: The number must be divisible by both 2 and 3.
6. Divisible by 7: There is no simple rule; typically, perform division or use trial and error.
7. Divisible by 8: The last three digits of the number must form a number divisible by 8.
8. Divisible by 9: The sum of the digits of the number must be divisible by 9.
9. Divisible by 10: The last digit must be 0.

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Step-by-Step Analysis:

#### Number: 995,368
- 2: Last digit is 8 (even) → Yes
- 3: Sum of digits = \(9 + 9 + 5 + 3 + 6 + 8 = 40\) (not divisible by 3) → No
- 4: Last two digits are 68 (68 ÷ 4 = 17) → Yes
- 5: Last digit is not 0 or 5 → No
- 6: Not divisible by 3 → No
- 7: Perform division: \(995,368 ÷ 7 = 142,195.4286\) (not an integer) → No
- 8: Last three digits are 368 (368 ÷ 8 = 46) → Yes
- 9: Sum of digits = 40 (not divisible by 9) → No
- 10: Last digit is not 0 → No

#### Number: 477
- 2: Last digit is 7 (odd) → No
- 3: Sum of digits = \(4 + 7 + 7 = 18\) (divisible by 3) → Yes
- 4: Last two digits are 77 (not divisible by 4) → No
- 5: Last digit is not 0 or 5 → No
- 6: Not divisible by 2 → No
- 7: Perform division: \(477 ÷ 7 = 68.1429\) (not an integer) → No
- 8: Last three digits are 477 (not divisible by 8) → No
- 9: Sum of digits = 18 (divisible by 9) → Yes
- 10: Last digit is not 0 → No

#### Number: 2,583
- 2: Last digit is 3 (odd) → No
- 3: Sum of digits = \(2 + 5 + 8 + 3 = 18\) (divisible by 3) → Yes
- 4: Last two digits are 83 (not divisible by 4) → No
- 5: Last digit is not 0 or 5 → No
- 6: Not divisible by 2 → No
- 7: Perform division: \(2,583 ÷ 7 = 369\) (integer) → Yes
- 8: Last three digits are 583 (not divisible by 8) → No
- 9: Sum of digits = 18 (divisible by 9) → Yes
- 10: Last digit is not 0 → No

#### Number: 70,688
- 2: Last digit is 8 (even) → Yes
- 3: Sum of digits = \(7 + 0 + 6 + 8 + 8 = 29\) (not divisible by 3) → No
- 4: Last two digits are 88 (88 ÷ 4 = 22) → Yes
- 5: Last digit is not 0 or 5 → No
- 6: Not divisible by 3 → No
- 7: Perform division: \(70,688 ÷ 7 = 10,098.2857\) (not an integer) → No
- 8: Last three digits are 688 (688 ÷ 8 = 86) → Yes
- 9: Sum of digits = 29 (not divisible by 9) → No
- 10: Last digit is not 0 → No

#### Number: 305,841
- 2: Last digit is 1 (odd) → No
- 3: Sum of digits = \(3 + 0 + 5 + 8 + 4 + 1 = 21\) (divisible by 3) → Yes
- 4: Last two digits are 41 (not divisible by 4) → No
- 5: Last digit is not 0 or 5 → No
- 6: Not divisible by 2 → No
- 7: Perform division: \(305,841 ÷ 7 = 43,691.5714\) (not an integer) → No
- 8: Last three digits are 841 (not divisible by 8) → No
- 9: Sum of digits = 21 (not divisible by 9) → No
- 10: Last digit is not 0 → No

#### Number: 735
- 2: Last digit is 5 (odd) → No
- 3: Sum of digits = \(7 + 3 + 5 = 15\) (divisible by 3) → Yes
- 4: Last two digits are 35 (not divisible by 4) → No
- 5: Last digit is 5 → Yes
- 6: Not divisible by 2 → No
- 7: Perform division: \(735 ÷ 7 = 105\) (integer) → Yes
- 8: Last three digits are 735 (not divisible by 8) → No
- 9: Sum of digits = 15 (not divisible by 9) → No
- 10: Last digit is not 0 → No

#### Number: 6,696
- 2: Last digit is 6 (even) → Yes
- 3: Sum of digits = \(6 + 6 + 9 + 6 = 27\) (divisible by 3) → Yes
- 4: Last two digits are 96 (96 ÷ 4 = 24) → Yes
- 5: Last digit is not 0 or 5 → No
- 6: Divisible by both 2 and 3 → Yes
- 7: Perform division: \(6,696 ÷ 7 = 956.5714\) (not an integer) → No
- 8: Last three digits are 696 (696 ÷ 8 = 87) → Yes
- 9: Sum of digits = 27 (divisible by 9) → Yes
- 10: Last digit is not 0 → No

#### Number: 821,170
- 2: Last digit is 0 (even) → Yes
- 3: Sum of digits = \(8 + 2 + 1 + 1 + 7 + 0 = 19\) (not divisible by 3) → No
- 4: Last two digits are 70 (not divisible by 4) → No
- 5: Last digit is 0 → Yes
- 6: Not divisible by 3 → No
- 7: Perform division: \(821,170 ÷ 7 = 117,310\) (integer) → Yes
- 8: Last three digits are 170 (not divisible by 8) → No
- 9: Sum of digits = 19 (not divisible by 9) → No
- 10: Last digit is 0 → Yes

#### Number: 14,049
- 2: Last digit is 9 (odd) → No
- 3: Sum of digits = \(1 + 4 + 0 + 4 + 9 = 18\) (divisible by 3) → Yes
- 4: Last two digits are 49 (not divisible by 4) → No
- 5: Last digit is not 0 or 5 → No
- 6: Not divisible by 2 → No
- 7: Perform division: \(14,049 ÷ 7 = 2,007\) (integer) → Yes
- 8: Last three digits are 049 (not divisible by 8) → No
- 9: Sum of digits = 18 (divisible by 9) → Yes
- 10: Last digit is not 0 → No

#### Number: 3,438
- 2: Last digit is 8 (even) → Yes
- 3: Sum of digits = \(3 + 4 + 3 + 8 = 18\) (divisible by 3) → Yes
- 4: Last two digits are 38 (not divisible by 4) → No
- 5: Last digit is not 0 or 5 → No
- 6: Divisible by both 2 and 3 → Yes
- 7: Perform division: \(3,438 ÷ 7 = 491.1429\) (not an integer) → No
- 8: Last three digits are 438 (not divisible by 8) → No
- 9: Sum of digits = 18 (divisible by 9) → Yes
- 10: Last digit is not 0 → No

#### Number: 28,872
- 2: Last digit is 2 (even) → Yes
- 3: Sum of digits = \(2 + 8 + 8 + 7 + 2 = 27\) (divisible by 3) → Yes
- 4: Last two digits are 72 (72 ÷ 4 = 18) → Yes
- 5: Last digit is not 0 or 5 → No
- 6: Divisible by both 2 and 3 → Yes
- 7: Perform division: \(28,872 ÷ 7 = 4,124.5714\) (not an integer) → No
- 8: Last three digits are 872 (872 ÷ 8 = 109) → Yes
- 9: Sum of digits = 27 (divisible by 9) → Yes
- 10: Last digit is not 0 → No

#### Number: 820
- 2: Last digit is 0 (even) → Yes
- 3: Sum of digits = \(8 + 2 + 0 = 10\) (not divisible by 3) → No
- 4: Last two digits are 20 (20 ÷ 4 = 5) → Yes
- 5: Last digit is 0 → Yes
- 6: Not divisible by 3 → No
- 7: Perform division: \(820 ÷ 7 = 117.1429\) (not an integer) → No
- 8: Last three digits are 820 (820 ÷ 8 = 102.5) → No
- 9: Sum of digits = 10 (not divisible by 9) → No
- 10: Last digit is 0 → Yes

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Final Answer:

\[
\boxed{
\begin{array}{c|ccccccccc}
& 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\
\hline
995,368 & \checkmark & & \checkmark & & & & \checkmark & & \\
477 & & \checkmark & & & & & & \checkmark & \\
2,583 & & \checkmark & & & & \checkmark & \checkmark & & \\
70,688 & \checkmark & & \checkmark & & & & \checkmark & & \\
305,841 & & \checkmark & & & & & & & \\
735 & & \checkmark & & \checkmark & & \checkmark & & & \\
6,696 & \checkmark & \checkmark & \checkmark & & \checkmark & & \checkmark & \checkmark & \\
821,170 & \checkmark & & & \checkmark & & \checkmark & & & \checkmark \\
14,049 & & \checkmark & & & & \checkmark & & \checkmark & \\
3,438 & \checkmark & \checkmark & & & \checkmark & & & \checkmark & \\
28,872 & \checkmark & \checkmark & \checkmark & & \checkmark & & \checkmark & \checkmark & \\
820 & \checkmark & & \checkmark & \checkmark & & & & & \checkmark \\
\end{array}
}
\]