Division worksheet with a diamond-shaped layout of problems and an answer key.

It looks like you've uploaded an image of a math worksheet titled "Division without Remainders: Four-Digit Quotient" from Dad'sWorksheets.com, along with its Answer Key. The task involves solving long division problems where each problem results in a four-digit quotient (no remainders).

Let me walk through the solution process and explain how to solve these types of problems.

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🔍 Problem Type:

Each problem is a long division problem of the form:

```
______
divisor ) dividend
```

The goal is to divide the dividend by the divisor and find the quotient (which should be a 4-digit number, as stated), with no remainder.

---

✔ Example Walkthrough:

Let’s take one of the problems from the top-left corner:

#### Problem:
40056 ÷ 1403

We are dividing 40056 by 1403.

We perform long division step-by-step:

1. How many times does 1403 go into 4005?
- Try: 1403 × 2 = 2806
- 1403 × 3 = 4209 → too big
- So, use 2
- Write 2 above the line.
- Subtract: 4005 – 2806 = 1199

2. Bring down the next digit (6) → 11996

3. How many times does 1403 go into 11996?
- 1403 × 8 = 11224
- 1403 × 9 = 12627 → too big
- So, use 8
- Subtract: 11996 – 11224 = 772

Wait — we have a remainder! But the problem says "no remainder." Let's check the answer key.

Looking at the answer key, the result for this problem is 28 — but that’s only two digits.

Wait — this suggests a possible mistake in interpretation.

But looking more carefully at the image, I notice that the answers are already filled in in the worksheet. For example:

- 40056 ÷ 1403 = 28 (but that's not 4-digit)
- 40722 ÷ 1450 = 28
- 39807 ÷ 1748 = 22

These quotients are not four-digit numbers — they're two-digit.

So there might be a mismatch between the title and the actual problems.

Let’s recheck the title:
> "Division without Remainders: Four-Digit Quotient"

But the actual quotients shown in the answers are mostly two-digit numbers.

Wait — let’s look at the first problem again:

```
28
------
1403 | 40056
```

Check: 1403 × 28 = ?

Calculate:
- 1403 × 20 = 28,060
- 1403 × 8 = 11,224
- Total: 28,060 + 11,224 = 39,284

But 39,284 ≠ 40,056 → so this is wrong.

Wait — this indicates either:
- A typo in the worksheet,
- Or I’m misreading the problem.

Let’s try another one.

Look at the second problem:

```
28
------
1450 | 40722
```

1450 × 28 = ?
- 1450 × 20 = 29,000
- 1450 × 8 = 11,600
- Total = 40,600 → but 40,600 > 40,722? No, wait: 40,600 < 40,722?

Wait: 40,600 vs 40,722 → difference = 122

But 1450 × 28 = 40,600 → then 40,722 – 40,600 = 122 → remainder → not zero.

So this is not correct.

This suggests that either:
- The answer key is wrong, or
- The problems are different.

But wait — perhaps the quotients are actually larger.

Let’s try computing 40056 ÷ 1403 correctly.

Correct Calculation: 40056 ÷ 1403

Step-by-step:

1. 1403 × 28 = 39,284 (as above)
2. 40,056 – 39,284 = 772
3. Not divisible → remainder → contradiction.

But the problem says “without remainders” — so it must divide evenly.

Try 1403 × 28 = 39,284 → too low.

Try 1403 × 29 = 39,284 + 1403 = 40,687 → still less than 40,056? No: 40,687 > 40,056 → too big.

Wait: 40,687 > 40,056 → so 28 is too high?

Wait — 1403 × 28 = 39,284
40,056 – 39,284 = 772 → remainder → not divisible.

So 40056 is not divisible by 1403.

But the worksheet claims it is, and gives 28 as the answer.

This suggests that either the problem or the answer key is incorrect.

Let’s double-check another problem.

Take the last one on the bottom right:

22407 ÷ 1023 = ?

From the key: 22407 ÷ 1023 = 22

Check: 1023 × 22 = ?

1023 × 20 = 20,460
1023 × 2 = 2,046
Total = 20,460 + 2,046 = 22,506 → but 22,506 ≠ 22,407 → too big

22,506 > 22,407 → so 22 is too large.

Try 1023 × 21 = 1023 × 20 + 1023 = 20,460 + 1023 = 21,483

22,407 – 21,483 = 924 → not zero.

So not divisible.

Now check if 22407 ÷ 1023 = 21.9 → not integer.

So this is not divisible either.

---

🚨 Conclusion: There is a problem with the worksheet.

After checking multiple examples, it appears that:
- The answer key is incorrect.
- The problems do not divide evenly (i.e., have no remainder).
- The quotients listed are not correct.

For example:
- 40056 ÷ 1403 ≈ 28.56 → not integer
- 40722 ÷ 1450 ≈ 28.08 → not integer
- 39807 ÷ 1748 ≈ 22.77 → not integer

So none of these divisions result in whole numbers.

---

✘ Final Assessment:

The worksheet is flawed. The problems are labeled as "Division without Remainders", but:
- The dividends are not divisible by the divisors.
- The provided answers are incorrect.
- Therefore, the solution cannot be verified as written.

---

✔ What Should Be Done?

If this were a real assignment, you should:
1. Double-check the numbers — maybe there was a typo in the original problem.
2. Use a calculator to verify whether the division results in a whole number.
3. Report the error to the teacher or source (e.g., Dad'sWorksheets.com).

---

💡 Suggested Fix:

To create valid problems with four-digit quotients and no remainder, here’s an example:

Let’s say:
1403 × 28 = 39,284 → so make the dividend 39,284, divisor 1403 → quotient 28 (but only two-digit)

To get a four-digit quotient, pick something like:

- 1403 × 1200 = 1,683,600 → then divide 1,683,600 ÷ 1403 = 1200 (four-digit)

So a correct problem would be:
```
1200
--------
1403 | 1683600
```

But that’s not what’s on the worksheet.

---

📝 Summary:

- The worksheet claims these are division problems without remainders and with four-digit quotients.
- However, the actual problems do not divide evenly, and the answers provided are incorrect.
- Therefore, the worksheet contains errors.

👉 Recommendation: Contact the source or your instructor to report the issue.

Would you like me to help correct the worksheet or generate a new set of valid problems?