Question

1. \\(7 \overline{)10,520}\\)

Explanation:

Step1: Divide the first digit

Divide 10 by 7. 7 goes into 10 once (1×7 = 7). Subtract 7 from 10, we get 3.
$$10 - 7 = 3$$

Step2: Bring down the next digit

Bring down the 5 to make 35. 7 goes into 35 exactly 5 times (5×7 = 35). Subtract 35 from 35, we get 0.
$$35 - 35 = 0$$

Step3: Bring down the next digit

Bring down the 2 to make 02 (which is 2). 7 goes into 2 zero times. Write 0 above the 2.

Step4: Bring down the last digit

Bring down the 0 to make 20. 7 goes into 20 two times (2×7 = 14). Subtract 14 from 20, we get 6.
$$20 - 14 = 6$$

Step5: Combine the results

Putting it all together, the quotient is 1.502 (with a remainder of 6, but since we're dealing with decimals, we can continue, but for the main quotient part: 1 (from 10) + 0.5 (from 35) + 0.00 (from 2) + 0.02 (from 20) = 1.528... Wait, actually, let's do long division properly:

10.520 ÷ 7:

- 7 into 10 (tens place) is 1, 1×7 = 7, 10 - 7 = 3.
- Bring down 5 (hundredths? Wait, 10.520 is 10 + 0.5 + 0.02 + 0.000. Wait, no, the number is 10.520, so the digits are 1 (ten), 0 (one),., 5 (tenth), 2 (hundredth), 0 (thousandth). Wait, maybe better to write as 10520 ÷ 7 (treating the decimal as 10.520 = 10520 × 10⁻³, but long division:

1.5028...
__________
7 | 10.5200
7
---
35
35
---
20
14
---
60
56
---
40
35
---
5...

Wait, maybe I made a mistake earlier. Let's do it step by step for 10.520 ÷ 7:

1. Divide 10 by 7: 1, remainder 3. Write 1 above the 0 (units place of 10.520).
2. Bring down the 5 (tenths place) to make 35. 35 ÷ 7 = 5. Write 5 above the 5 (tenths place). Now we have 1.5.
3. Bring down the 2 (hundredths place) to make 02 (since 35 - 35 = 0, bring down 2). 2 ÷ 7 = 0, write 0 above the 2 (hundredths place). Now we have 1.50.
4. Bring down the 0 (thousandths place) to make 20. 20 ÷ 7 = 2 (2×7=14), remainder 6. Write 2 above the 0 (thousandths place). Now we have 1.502.
5. If we want to continue, bring down a 0 (ten - thousandths place) to make 60. 60 ÷ 7 = 8 (8×7=56), remainder 4. Write 8 above the new 0. So now 1.5028...

But the main quotient up to the thousandths place is 1.502 (with remainder 6, but for the division of 10.520 by 7, the result is 1.502857... approximately. But maybe the question is to do the long division of 10520 ÷ 7 (treating 10.520 as 10520 × 10⁻³, so 10520 ÷ 7 = 1502.857..., then × 10⁻³ = 1.502857...

Wait, maybe the original number is 10,520 (ten thousand five hundred twenty) divided by 7? But the notation is 7√10,520, which is 10,520 ÷ 7. Let's do that:

10,520 ÷ 7:

- 7 into 10 (thousands place) is 1 (1×7=7), remainder 3.
- Bring down 5 (hundreds place) to make 35. 35 ÷ 7 = 5, remainder 0.
- Bring down 2 (tens place) to make 02 (which is 2). 2 ÷ 7 = 0, remainder 2.
- Bring down 0 (ones place) to make 20. 20 ÷ 7 = 2, remainder 6.

So 10,520 ÷ 7 = 1502 with a remainder of 6. So as a decimal, 1502.857... (since 6/7 ≈ 0.857).

Wait, I think I misread the number. The number is 10,520 (ten thousand five hundred twenty) or 10.520 (ten point five two zero). The division symbol is 7√10,520, which is 10,520 ÷ 7. Let's confirm:

Long division for 10520 ÷ 7:

1502
_______
7 | 10520
7
---
35
35
---
20
14
---
6

Wait, no: 7×1502 = 10514, 10520 - 10514 = 6. So quotient is 1502, remainder 6. If it's 10.520 ÷ 7, then:

10.520 ÷ 7 = (10 + 0.5 + 0.02 + 0.000) ÷ 7 = 10÷7 + 0.5÷7 + 0.02÷7 + 0.000÷7 = 1.428... + 0.071... + 0.0028... + 0 = 1.5028...

Ah, I see, the comma in 10,520 is a thousand separator, so it's 10520 (ten thousand five hundred twenty) or 10.520 (ten point five two zero). The division is 10.520 ÷ 7 (since the decimal is present). So let's do that:

1. 7 into 10 (the 10 before the decimal) is 1, 1×7=7, 10-7=3.
2. Bring down the 5 (after the decimal) to make 35 (now 3.5, but in decimal terms, 3.5 ÷ 7 = 0.5). So write 0.5, so now we have 1.5.
3. Bring down the 2: now we have 0.02 (since after 3.5, we have 0.02 left? Wait, no, 10.520 is 10 + 0.5 + 0.02 + 0.000. So after 10÷7=1.428..., then 0.5÷7=0.071..., 0.02÷7=0.0028..., so total is 1.428...+0.071...+0.0028...=1.5028...

But the long division steps:

1.5028...
__________
7 | 10.5200
7
---
35
35
---
20
14
---
60
56
---
40
35
---
5...

So the quotient is approximately 1.5029 (rounded). But maybe the question is to do 10520 ÷ 7 (the comma as thousand separator, so 10,520 = 10520). Then 10520 ÷ 7:

7×1500=10500, 10520-10500=20, 20÷7=2 with remainder 6. So 1500+2=1502, remainder 6. So 1502 6/7 ≈ 1502.857.

But the original problem has a decimal, so it's 10.520 ÷ 7. Let's do that properly:

10.520 ÷ 7:

- 7 goes into 10 once (1), 1×7=7, 10-7=3.
- Bring down the 5 to make 35 (now we're in the tenths place), 35÷7=5, so write 5 in the tenths place. Now we have 1.5.
- Bring down the 2 to make 02 (hundredths place), 2÷7=0, write 0 in the hundredths place.
- Bring down the 0 to make 20 (thousandths place), 20÷7=2 (2×7=14), write 2 in the thousandths place, 20-14=6.
- Now we can bring down another 0 to make 60 (ten - thousandths place), 60÷7=8 (8×7=56), 60-56=4.
- Bring down another 0 to make 40 (hundred - thousandths place), 40÷7=5 (5×7=35), 40-35=5.
- And so on.

So the quotient is 1.502857..., which can be written as 1.5029 (rounded to four decimal places) or 1.503 (rounded to three). But the main long division result for the given number (10.520 ÷ 7) is 1.502857...

Answer:

The result of \( 10.520 \div 7 \) (or \( 10,520 \div 7 \) if it's a whole number, but with the decimal, it's approximately) \( 1.5029 \) (or exactly \( 1 \frac{36}{70} = 1 \frac{18}{35} \approx 1.514 \)? Wait, no, my earlier steps were confused. Let's do 10.520 ÷ 7:

10.520 ÷ 7 = (10 + 0.5 + 0.02 + 0.000) ÷ 7 = 10/7 + 0.5/7 + 0.02/7 + 0.000/7 = 1.428571... + 0.071428... + 0.002857... + 0 = 1.502857...

Yes, so the quotient is approximately 1.5029. If we do long division for 10.520 ÷ 7:

    1.502857...
  ______________
7 | 10.520000...
    7
    ---
    35
    35
    ---
      20
      14
      ---
       60
       56
       ---
        40
        35
        ---
         50
         49
         ---
          10
          7
          ---
           3... (and it repeats)

So the final answer, when dividing 10.520 by 7, is \( \boldsymbol{1.502857\ldots} \) (or \( 1.503 \) when rounded to three decimal places).