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To solve these problems, we use the Divisibility Rule for 3:

> A number is divisible by 3 if the sum of its digits is divisible by 3.

Let’s go step by step.

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Part A: State whether the numbers are divisible by 3.

We’ll add the digits of each number and check if the sum is divisible by 3.

1) 1515
Digits: 1 + 5 + 1 + 5 = 12
12 ÷ 3 = 4 → No remainder → ✔ Divisible by 3

2) 7747
Digits: 7 + 7 + 4 + 7 = 25
25 ÷ 3 = 8 with remainder 1 → ✘ Not divisible by 3

3) 656
Digits: 6 + 5 + 6 = 17
17 ÷ 3 = 5 with remainder 2 → ✘ Not divisible by 3

4) 654
Digits: 6 + 5 + 4 = 15
15 ÷ 3 = 5 → No remainder → ✔ Divisible by 3

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Part B: Which of the following numbers is divisible by 3?

Check each option:

a) 450 → 4 + 5 + 0 = 9 → 9 ÷ 3 = 3 → ✔ Yes
b) 678 → 6 + 7 + 8 = 21 → 21 ÷ 3 = 7 → ✔ Yes
c) 630 → 6 + 3 + 0 = 9 → 9 ÷ 3 = 3 → ✔ Yes
d) 780 → 7 + 8 + 0 = 15 → 15 ÷ 3 = 5 → ✔ Yes

Wait — all four are divisible by 3? That seems odd for a multiple-choice question. Let me double-check:

- 450: 4+5+0=9 → yes
- 678: 6+7+8=21 → yes
- 630: 6+3+0=9 → yes
- 780: 7+8+0=15 → yes

Hmm… maybe the question expects us to pick all that apply, but it says “which” (singular). But since all are divisible, perhaps there's a trick? Or maybe it’s designed to show that more than one can be correct? But in standard tests, sometimes they expect you to pick one — but here, all work.

Actually, looking again — maybe I misread? The question says: “Which of the following numbers is divisible by 3?” — and all four are. So perhaps it’s a trick question or allows multiple answers. But since it’s listed as a-b-c-d, maybe it’s expecting any one? But that doesn’t make sense.

Wait — let me check if I made a mistake.

678: 6+7+8=21 → 21÷3=7 → correct
780: 7+8+0=15 → correct
All are correct.

But perhaps the worksheet has a typo? Or maybe in context, only one is intended? Since all are divisible, I’ll note that all are divisible — but if forced to choose one, any would work. However, since this is likely meant to have one answer, let me recheck the original image description — no, the user didn’t specify.

Actually, looking back at the problem statement: it says “Which of the following numbers is divisible by 3?” — and lists a,b,c,d. In many worksheets, even if multiple are correct, they might expect you to select all — but since it’s not specified, and given that all are divisible, perhaps the question is flawed. But for accuracy, I must state the truth.

However, let me see Part C — maybe that will help clarify.

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Part C: Which of the following numbers is not divisible by 3?

Check each:

a) 566 → 5 + 6 + 6 = 17 → 17 ÷ 3 = 5 R2 → ✘ Not divisible → This is a candidate
b) 977 → 9 + 7 + 7 = 23 → 23 ÷ 3 = 7 R2 → ✘ Not divisible → Also candidate
c) 609 → 6 + 0 + 9 = 15 → 15 ÷ 3 = 5 → ✔ Divisible
d) 545 → 5 + 4 + 5 = 14 → 14 ÷ 3 = 4 R2 → ✘ Not divisible

So a, b, and d are NOT divisible by 3. Again, multiple answers.

This suggests that perhaps the questions are designed to have multiple correct answers, or maybe I’m missing something.

Wait — let me re-read Part B: “Which of the following numbers is divisible by 3?” — and all four are. Similarly, Part C has three that are not.

But in standard educational materials, sometimes they design questions where only one is correct. Let me verify my calculations once more.

Part B:
- 450: 4+5+0=9 → divisible by 3 → yes
- 678: 6+7+8=21 → yes
- 630: 6+3+0=9 → yes
- 780: 7+8+0=15 → yes

All correct.

Part C:
- 566: 5+6+6=17 → not divisible
- 977: 9+7+7=23 → not divisible
- 609: 6+0+9=15 → divisible → so this is the only one that IS divisible, meaning the others are not
- 545: 5+4+5=14 → not divisible

So for Part C, the numbers that are NOT divisible are a, b, d. But the question asks “which... is not divisible” — implying one answer. But there are three.

This is confusing. Perhaps there’s a mistake in the worksheet, or perhaps I need to select all that apply. But since the instruction is to solve accurately, I will provide the correct mathematical answers.

For Part B, since all are divisible, any of them could be chosen, but typically in such cases, if it’s single choice, it might be a error. However, looking at common worksheets, sometimes they include distractors, but here all work.

Another thought: maybe for Part B, it’s “which one” and they expect you to pick one, but since all are correct, perhaps it’s fine. But to be precise, I’ll list the correct responses.

Perhaps the worksheet intends for students to identify that all are divisible in B, and for C, to identify that c is divisible, so the others are not — but the question is phrased as “which is not”, so a,b,d are correct.

But since the final answer format requires a specific output, and given that this is likely a standard worksheet, I recall that in some versions, Part B might have a different set, but based on what’s given, I must go with the math.

Let me assume that for Part B, since all are divisible, but perhaps the question is to select one, and I can choose any. But that’s not satisfactory.

Wait — let me check online or think differently. Perhaps I miscalculated 678? 6+7+8=21, yes. 780: 7+8+0=15, yes. All good.

Perhaps the answer for Part B is all of them, but since it’s multiple choice with single letters, maybe it’s designed to have only one, but in this case, it’s not.

Another idea: maybe "divisible by 3" means exactly divisible with no remainder, which all are, so all are correct.

For the sake of completing the task, I'll provide the answers as per calculation.

For Part A:
1) Yes
2) No
3) No
4) Yes

For Part B: All are divisible, so a, b, c, d are all correct. But since it's likely expecting one, and perhaps in the original context, but I think I should state the truth.

Perhaps the worksheet has a typo, but based on given numbers, all in B are divisible.

For Part C: a, b, d are not divisible, c is divisible. So the numbers that are not divisible are a, b, d.

But the question says "which... is not divisible", so any of a,b,d could be selected, but again, multiple.

This is problematic for a final answer.

Let me look back at the user's request: "Solve the problem accurately." So I must give the correct mathematical answers.

Perhaps for Part B, since it's "which", and all are, but in some contexts, they might expect the first one, but that's arbitrary.

I recall that in some worksheets, for Part B, they might have intended only one to be divisible, but here all are. Let me double-check 678: 678 ÷ 3 = 226, yes. 780 ÷ 3 = 260, yes. All good.

Perhaps the answer for Part B is that all are divisible, but since the format is to choose one letter, and the instruction is to provide final answer, I need to see how to box it.

Another thought: maybe for Part B, it's a trick, and they want us to see that all are, but typically, in such cases, the answer might be "all of the above" but it's not listed.

Given that, and to move forward, I'll provide the answers as per calculation, and for the final answer, I'll list them as required.

Perhaps the user expects the answers for each part.

Let me structure the final answer as:

For A: 1) Yes, 2) No, 3) No, 4) Yes

For B: All are divisible, so a, b, c, d

For C: a, b, d are not divisible

But since the final answer section might expect a specific format, and the instruction is to provide only the final correct answer, I need to see.

Perhaps for each subpart, but the user didn't specify.

Looking at the response format, it says "Final Answer: [Provide only the final correct answer]"

So perhaps I need to compile all.

But to be practical, and since this is a worksheet, I'll assume that for Part B, even though all are divisible, perhaps the intended answer is to recognize that, but for the sake of the exercise, I'll provide the calculations.

I think I found a way: in Part B, perhaps the question is "which" and in some versions, only one is correct, but here all are, so I'll state that all are divisible.

Similarly for C, three are not.

But for the final answer, I'll write:

Part A: 1) Yes, 2) No, 3) No, 4) Yes
Part B: a, b, c, d (all are divisible)
Part C: a, b, d (not divisible)

But to match the format, perhaps list the answers.

Since the user might expect a single boxed answer, but there are multiple parts, I think I need to provide the answers for each.

Perhaps the final answer is the completion of the worksheet.

Let me try to write it as:

For A:
1) Yes
2) No
3) No
4) Yes

For B: All options (a, b, c, d) are divisible by 3.
For C: Options a, b, and d are not divisible by 3.

But for the boxed final answer, perhaps I can write the key points.

Given the constraints, I'll provide the final answer as the solutions to each part.

So, after careful calculation:

Final Answer:

A) 1) Yes, 2) No, 3) No, 4) Yes
B) All of the above (a, b, c, d) are divisible by 3.
C) a, b, and d are not divisible by 3.

But to make it neat, and since the instruction is to box the final answer, I'll write it as:

\boxed{\text{A: 1) Yes, 2) No, 3) No, 4) Yes; B: a,b,c,d; C: a,b,d}}

But let's confirm Part C: the question is "which is not divisible", and a,b,d are not, c is, so yes.

For Part B, since all are divisible, and if the question allows multiple, then a,b,c,d.

I think that's accurate.

So, to summarize:

- For A, we have yes/no for each.
- For B, all options work.
- For C, a,b,d do not work.

So the final answer is as above.