Divisibility rules worksheet for numbers 3, 6, and 9 with practice problems.

To solve the problem, we need to apply the divisibility rules for 3, 6, and 9. Let's go through each section step by step.

Divisibility Rules Recap:

1. Divisible by 3: The sum of the digits of the number must be divisible by 3.
2. Divisible by 6: The number must be divisible by both 2 and 3.
- Divisible by 2: The last digit must be even (0, 2, 4, 6, or 8).
- Divisible by 3: The sum of the digits must be divisible by 3.
3. Divisible by 9: The sum of the digits of the number must be divisible by 9.

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Section 1: Divisible by 3?

We check if the sum of the digits of each number is divisible by 3.

- 562: \(5 + 6 + 2 = 13\) (not divisible by 3)
- 491: \(4 + 9 + 1 = 14\) (not divisible by 3)
- 187: \(1 + 8 + 7 = 16\) (not divisible by 3)
- 702: \(7 + 0 + 2 = 9\) (divisible by 3)
- 360: \(3 + 6 + 0 = 9\) (divisible by 3)
- 427: \(4 + 2 + 7 = 13\) (not divisible by 3)
- 344: \(3 + 4 + 4 = 11\) (not divisible by 3)
- 965: \(9 + 6 + 5 = 20\) (not divisible by 3)
- 444: \(4 + 4 + 4 = 12\) (divisible by 3)
- 127: \(1 + 2 + 7 = 10\) (not divisible by 3)
- 185: \(1 + 8 + 5 = 14\) (not divisible by 3)
- 814: \(8 + 1 + 4 = 13\) (not divisible by 3)
- 933: \(9 + 3 + 3 = 15\) (divisible by 3)
- 458: \(4 + 5 + 8 = 17\) (not divisible by 3)
- 799: \(7 + 9 + 9 = 25\) (not divisible by 3)
- 847: \(8 + 4 + 7 = 19\) (not divisible by 3)
- 432: \(4 + 3 + 2 = 9\) (divisible by 3)
- 355: \(3 + 5 + 5 = 13\) (not divisible by 3)
- 760: \(7 + 6 + 0 = 13\) (not divisible by 3)
- 763: \(7 + 6 + 3 = 16\) (not divisible by 3)
- 241: \(2 + 4 + 1 = 7\) (not divisible by 3)
- 477: \(4 + 7 + 7 = 18\) (divisible by 3)
- 139: \(1 + 3 + 9 = 13\) (not divisible by 3)
- 640: \(6 + 4 + 0 = 10\) (not divisible by 3)

Numbers divisible by 3:
702, 360, 444, 933, 432, 477

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Section 2: Divisible by 6?

A number is divisible by 6 if it is divisible by both 2 and 3.

- 773: Not divisible by 2 (last digit is odd).
- 701: Not divisible by 2 (last digit is odd).
- 553: Not divisible by 2 (last digit is odd).
- 684: Divisible by 2 (last digit is 4). Sum of digits: \(6 + 8 + 4 = 18\) (divisible by 3). Yes.
- 162: Divisible by 2 (last digit is 2). Sum of digits: \(1 + 6 + 2 = 9\) (divisible by 3). Yes.
- 501: Not divisible by 2 (last digit is odd).
- 619: Not divisible by 2 (last digit is odd).
- 985: Not divisible by 2 (last digit is odd).
- 776: Divisible by 2 (last digit is 6). Sum of digits: \(7 + 7 + 6 = 20\) (not divisible by 3).
- 816: Divisible by 2 (last digit is 6). Sum of digits: \(8 + 1 + 6 = 15\) (divisible by 3). Yes.
- 883: Not divisible by 2 (last digit is odd).
- 212: Divisible by 2 (last digit is 2). Sum of digits: \(2 + 1 + 2 = 5\) (not divisible by 3).
- 934: Divisible by 2 (last digit is 4). Sum of digits: \(9 + 3 + 4 = 16\) (not divisible by 3).
- 295: Not divisible by 2 (last digit is odd).
- 466: Divisible by 2 (last digit is 6). Sum of digits: \(4 + 6 + 6 = 16\) (not divisible by 3).
- 811: Not divisible by 2 (last digit is odd).
- 431: Not divisible by 2 (last digit is odd).
- 945: Divisible by 2 (last digit is 5). Sum of digits: \(9 + 4 + 5 = 18\) (divisible by 3). Yes.
- 143: Not divisible by 2 (last digit is odd).
- 336: Divisible by 2 (last digit is 6). Sum of digits: \(3 + 3 + 6 = 12\) (divisible by 3). Yes.
- 861: Not divisible by 2 (last digit is odd).
- 526: Divisible by 2 (last digit is 6). Sum of digits: \(5 + 2 + 6 = 13\) (not divisible by 3).
- 864: Divisible by 2 (last digit is 4). Sum of digits: \(8 + 6 + 4 = 18\) (divisible by 3). Yes.
- 909: Not divisible by 2 (last digit is odd).

Numbers divisible by 6:
684, 162, 816, 945, 336, 864

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Section 3: Divisible by 9?

We check if the sum of the digits of each number is divisible by 9.

- 986: \(9 + 8 + 6 = 23\) (not divisible by 9)
- 386: \(3 + 8 + 6 = 17\) (not divisible by 9)
- 342: \(3 + 4 + 2 = 9\) (divisible by 9)
- 143: \(1 + 4 + 3 = 8\) (not divisible by 9)
- 350: \(3 + 5 + 0 = 8\) (not divisible by 9)
- 201: \(2 + 0 + 1 = 3\) (not divisible by 9)
- 611: \(6 + 1 + 1 = 8\) (not divisible by 9)
- 368: \(3 + 6 + 8 = 17\) (not divisible by 9)
- 267: \(2 + 6 + 7 = 15\) (not divisible by 9)
- 160: \(1 + 6 + 0 = 7\) (not divisible by 9)
- 168: \(1 + 6 + 8 = 15\) (not divisible by 9)
- 904: \(9 + 0 + 4 = 13\) (not divisible by 9)
- 919: \(9 + 1 + 9 = 19\) (not divisible by 9)
- 201: \(2 + 0 + 1 = 3\) (not divisible by 9)
- 314: \(3 + 1 + 4 = 8\) (not divisible by 9)
- 545: \(5 + 4 + 5 = 14\) (not divisible by 9)
- 621: \(6 + 2 + 1 = 9\) (divisible by 9)
- 741: \(7 + 4 + 1 = 12\) (not divisible by 9)
- 579: \(5 + 7 + 9 = 21\) (not divisible by 9)
- 851: \(8 + 5 + 1 = 14\) (not divisible by 9)
- 326: \(3 + 2 + 6 = 11\) (not divisible by 9)
- 501: \(5 + 0 + 1 = 6\) (not divisible by 9)
- 422: \(4 + 2 + 2 = 8\) (not divisible by 9)
- 577: \(5 + 7 + 7 = 19\) (not divisible by 9)

Numbers divisible by 9:
342, 621

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Section 4: Divisible by 3 and 6?

From Section 2, we already identified numbers divisible by 6. Since all numbers divisible by 6 are also divisible by 3, the answer is the same as Section 2.

Numbers divisible by 3 and 6:
684, 162, 816, 945, 336, 864

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Section 5: Divisible by 3 and 9?

A number divisible by both 3 and 9 must be divisible by 9. From Section 3, we already identified numbers divisible by 9.

Numbers divisible by 3 and 9:
342, 621

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Final Answer:

\[
\boxed{
\begin{array}{l}
\text{Divisible by 3: } 702, 360, 444, 933, 432, 477 \\
\text{Divisible by 6: } 684, 162, 816, 945, 336, 864 \\
\text{Divisible by 9: } 342, 621 \\
\text{Divisible by 3 and 6: } 684, 162, 816, 945, 336, 864 \\
\text{Divisible by 3 and 9: } 342, 621
\end{array}
}
\]