Math worksheet for practicing divisibility rules for 3, 6, and 9.

To solve the problem, we need to determine which numbers are divisible by 3, 6, and 9. Let's break it down step by step.

Divisibility Rules:

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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Step 1: Numbers Divisible by 3

We will check each number by summing its digits and verifying if the sum is divisible by 3.

#### Row 1:
- 1,128: \(1 + 1 + 2 + 8 = 12\) (divisible by 3) → Circle
- 2,101: \(2 + 1 + 0 + 1 = 4\) (not divisible by 3)
- 2,235: \(2 + 2 + 3 + 5 = 12\) (divisible by 3) → Circle
- 3,122: \(3 + 1 + 2 + 2 = 8\) (not divisible by 3)
- 4,445: \(4 + 4 + 4 + 5 = 17\) (not divisible by 3)
- 2,088: \(2 + 0 + 8 + 8 = 18\) (divisible by 3) → Circle

#### Row 2:
- 9,457: \(9 + 4 + 5 + 7 = 25\) (not divisible by 3)
- 5,004: \(5 + 0 + 0 + 4 = 9\) (divisible by 3) → Circle
- 6,625: \(6 + 6 + 2 + 5 = 19\) (not divisible by 3)
- 7,134: \(7 + 1 + 3 + 4 = 15\) (divisible by 3) → Circle
- 9,120: \(9 + 1 + 2 + 0 = 12\) (divisible by 3) → Circle
- 5,228: \(5 + 2 + 2 + 8 = 17\) (not divisible by 3)

#### Row 3:
- 4,002: \(4 + 0 + 0 + 2 = 6\) (divisible by 3) → Circle
- 1,124: \(1 + 1 + 2 + 4 = 8\) (not divisible by 3)
- 7,886: \(7 + 8 + 8 + 6 = 29\) (not divisible by 3)
- 9,999: \(9 + 9 + 9 + 9 = 36\) (divisible by 3) → Circle
- 1,750: \(1 + 7 + 5 + 0 = 13\) (not divisible by 3)
- 5,661: \(5 + 6 + 6 + 1 = 18\) (divisible by 3) → Circle

Numbers divisible by 3:
- 1,128, 2,235, 2,088, 5,004, 7,134, 9,120, 4,002, 9,999, 5,661

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Step 2: Numbers Divisible by 6

We will check each number to ensure it is divisible by both 2 and 3.

#### Row 1:
- 7,500: Last digit is 0 (even), sum of digits \(7 + 5 + 0 + 0 = 12\) (divisible by 3) → Circle
- 9,567: Last digit is 7 (not even)
- 9,072: Last digit is 2 (even), sum of digits \(9 + 0 + 7 + 2 = 18\) (divisible by 3) → Circle
- 2,199: Last digit is 9 (not even)
- 6,864: Last digit is 4 (even), sum of digits \(6 + 8 + 6 + 4 = 24\) (divisible by 3) → Circle
- 5,400: Last digit is 0 (even), sum of digits \(5 + 4 + 0 + 0 = 9\) (divisible by 3) → Circle

#### Row 2:
- 5,290: Last digit is 0 (even), sum of digits \(5 + 2 + 9 + 0 = 16\) (not divisible by 3)
- 7,398: Last digit is 8 (even), sum of digits \(7 + 3 + 9 + 8 = 27\) (divisible by 3) → Circle
- 6,092: Last digit is 2 (even), sum of digits \(6 + 0 + 9 + 2 = 17\) (not divisible by 3)
- 9,666: Last digit is 6 (even), sum of digits \(9 + 6 + 6 + 6 = 27\) (divisible by 3) → Circle
- 8,862: Last digit is 2 (even), sum of digits \(8 + 8 + 6 + 2 = 24\) (divisible by 3) → Circle
- 3,125: Last digit is 5 (not even)

#### Row 3:
- 6,834: Last digit is 4 (even), sum of digits \(6 + 8 + 3 + 4 = 21\) (divisible by 3) → Circle
- 2,907: Last digit is 7 (not even)
- 5,335: Last digit is 5 (not even)
- 7,421: Last digit is 1 (not even)
- 3,444: Last digit is 4 (even), sum of digits \(3 + 4 + 4 + 4 = 15\) (divisible by 3) → Circle
- 7,728: Last digit is 8 (even), sum of digits \(7 + 7 + 2 + 8 = 24\) (divisible by 3) → Circle

Numbers divisible by 6:
- 7,500, 9,072, 6,864, 5,400, 7,398, 9,666, 8,862, 6,834, 3,444, 7,728

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Step 3: Numbers Divisible by 9

We will check each number by summing its digits and verifying if the sum is divisible by 9.

#### Row 1:
- 9,099: \(9 + 0 + 9 + 9 = 27\) (divisible by 9) → Circle
- 1,145: \(1 + 1 + 4 + 5 = 11\) (not divisible by 9)
- 8,089: \(8 + 0 + 8 + 9 = 25\) (not divisible by 9)
- 6,228: \(6 + 2 + 2 + 8 = 18\) (divisible by 9) → Circle
- 9,139: \(9 + 1 + 3 + 9 = 22\) (not divisible by 9)
- 4,950: \(4 + 9 + 5 + 0 = 18\) (divisible by 9) → Circle

#### Row 2:
- 6,192: \(6 + 1 + 9 + 2 = 18\) (divisible by 9) → Circle
- 4,383: \(4 + 3 + 8 + 3 = 18\) (divisible by 9) → Circle
- 4,188: \(4 + 1 + 8 + 8 = 21\) (not divisible by 9)
- 1,099: \(1 + 0 + 9 + 9 = 19\) (not divisible by 9)
- 7,144: \(7 + 1 + 4 + 4 = 16\) (not divisible by 9)
- 1,926: \(1 + 9 + 2 + 6 = 18\) (divisible by 9) → Circle

#### Row 3:
- 8,865: \(8 + 8 + 6 + 5 = 27\) (divisible by 9) → Circle
- 9,455: \(9 + 4 + 5 + 5 = 23\) (not divisible by 9)
- 3,322: \(3 + 3 + 2 + 2 = 10\) (not divisible by 9)
- 2,997: \(2 + 9 + 9 + 7 = 27\) (divisible by 9) → Circle
- 1,035: \(1 + 0 + 3 + 5 = 9\) (divisible by 9) → Circle
- 1,494: \(1 + 4 + 9 + 4 = 18\) (divisible by 9) → Circle

Numbers divisible by 9:
- 9,099, 6,228, 4,950, 6,192, 4,383, 1,926, 8,865, 2,997, 1,035, 1,494

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Final Answer:

1. Divisible by 3:
\[
\boxed{1,128, 2,235, 2,088, 5,004, 7,134, 9,120, 4,002, 9,999, 5,661}
\]

2. Divisible by 6:
\[
\boxed{7,500, 9,072, 6,864, 5,400, 7,398, 9,666, 8,862, 6,834, 3,444, 7,728}
\]

3. Divisible by 9:
\[
\boxed{9,099, 6,228, 4,950, 6,192, 4,383, 1,926, 8,865, 2,997, 1,035, 1,494}
\]