Long Division: Division by a 2-digit number

To solve the division problem $\frac{648}{12}$, we'll use the long division method. Here's each step broken down:

• Step 1: Begin by writing 648 under the division bar and 12 outside the bar.
• Step 2: Determine how many times 12 fits into the first digit, 6. Since 12 is greater than 6, check the first two digits together, 64.
• Step 3: Determine how many times 12 goes into 64. The answer is 5, because $12 \times 5 = 60$, which is less than 64. Write 5 above the division bar.
• Step 4: Subtract 60 from 64, leaving a remainder of 4. Carry down the next digit, 8, to get 48.
• Step 5: Determine how many times 12 goes into 48. The answer is 4, since $12 \times 4 = 48$.
• Step 6: Subtract 48 from 48, leaving a remainder of 0.

The quotient of $\frac{648}{12}$ is, therefore, 54.

Thus, the solution to the problem is $54$.

To solve the problem of finding $945 \div 15$, we will use long division:

• Step 1: Set up the long division. With $945$ as the dividend and $15$ as the divisor, begin by considering the first two digits, $94$.

• Step 2: Determine how many times $15$ fits into $94$. Since $15 \times 6 = 90$ (the highest multiple of 15 under 94), write $6$ as the first digit of the quotient.

• Step 3: Subtract $90$ from $94$, obtaining $4$ (as $94 - 90 = 4$).

• Step 4: Bring down the next digit from the dividend, which is $5$, turning the current number to $45$.

• Step 5: Find how many times $15$ fits into $45$. Since $45 \div 15 = 3$ exactly, write $3$ as the next digit of the quotient.

• Step 6: Subtract $45$ from $45$, resulting in $0$. There are no remaining digits to bring down, and the process is complete.

Therefore, the result of $945 \div 15$ is $63$.

To solve this problem, we'll follow these steps to perform long division:

• Step 1: Divide 8 by 13. Thirteen goes into 8 zero times, so check next digit to make 84.
• Step 2: Divide 84 by 13. Thirteen fits into 84 six times (since $13 \times 6 = 78$).
• Step 3: Subtract the result from 84, giving $84 - 78 = 6$.
• Step 4: Bring down the next digit of 845, which is 5, making it 65.
• Step 5: Divide 65 by 13. Thirteen fits into 65 five times (since $13 \times 5 = 65$).
• Step 6: Subtract the result from 65, giving $65 - 65 = 0$.

The division ends here with no remainder, meaning our quotient is complete.

Therefore, the quotient of 845 divided by 13 is $65$.

To solve this problem, we'll conduct a long division of 325 by 14 step by step:

• Step 1: Determine how many times 14 fits into the first one or two digits of 325.
• Step 2: The first part of 325 we consider is 32 (as 14 is larger than 3). 14 fits into 32 two times because $14 \times 2 = 28$ and $14 \times 3 = 42$, which is too large. Write 2 as part of the quotient.
• Step 3: Subtract $28$ from $32$ to get $4$.
• Step 4: Bring down the next digit of the dividend, which is 5, forming 45.
• Step 5: Determine how many times 14 fits into 45. It fits three times, since $14 \times 3 = 42$ and $14 \times 4 = 56$ is too large. Write 3 as part of the quotient.
• Step 6: Subtract $42$ from $45$ to get $3$.
• Step 7: There are no more digits to bring down from the dividend, so the remainder is 3.

The quotient of $325 \div 14$ is $23$ with a remainder of $3$.

Therefore, the solution to the problem is $23$ with a remainder of $3$.

To solve this division problem of dividing 1140 by 15, we perform the following steps:

• Step 1: Set up the division with 1140 under the division bracket and 15 outside.
• Step 2: Determine how many times 15 fits into the first two digits of the dividend, 11. Since it doesn't fit, consider the first three digits, 114.
• Step 3: Calculate $15 \times 7 = 105$, which is the nearest lower product to 114. Write 7 above the division bar.
• Step 4: Subtract 105 from 114, leaving 9. Bring down the next digit of the dividend, 0, making it 90.
• Step 5: Determine how many times 15 fits into 90. $15 \times 6 = 90$. Write 6 above the division bar.
• Step 6: Subtract the 90 from 90, resulting in a remainder of 0.

The quotient from the division of 1140 by 15 is $76$. None remains, confirming complete division.

Therefore, the solution to the problem is $76$.

To solve this problem, we'll follow these steps using long division:

• Step 1: Set up the problem by writing 3240 under the division bracket and 16 outside.
• Step 2: Determine how many times 16 goes into the leading number within the dividend.

Let's work through each step:

Step 1: Consider the first two digits of 3240, which is 32. Divide 32 by 16.

Step 2: $16$ goes into $32$ two times. Write $2$ above the line as the first digit of the quotient.

Step 3: Multiply $2$ by $16$ to get $32$, and subtract this from the current digit segment, which is $32$. We have no remainder yet, so bring down the next digit, which is $4$.

Step 4: Now divide $40$ (after bringing down the next digit) by $16$. $16$ goes into $40$ two times as well. Write $2$ as the next digit of the quotient.

Step 5: Multiply $2$ by $16$ to get $32$, and subtract from the current digit segment, $40$. This gives a remainder of $8$, and we bring down the last $0$, which now makes $80$.

Step 6: $16$ goes into $80$ five times. Write $5$ as the last digit of the quotient.

Step 7: Multiply $5$ by $16$ to get $80$, subtract from $80$ to get a remainder of $0$. Bring down the final $0$, there is no digit remaining after this.

Therefore, the solution to the problem is that the quotient is $202$ with a remainder of $8$, which matches choice 4.

To solve the division problem of $3540 \div 17$, follow these steps:

• Step 1: Determine how many times 17 can fit into 35. It fits 2 times.
• Step 2: Multiply 17 by 2 to get 34, and subtract this from 35 to get a remainder of 1.
• Step 3: Bring down the next digit from 3540, which is 4, to make it 14.
• Step 4: 17 does not fit into 14, so put 0 in the quotient.
• Step 5: Bring down the next digit, which is 0, to make 140.
• Step 6: Determine how many times 17 fits into 140. It fits 8 times.
• Step 7: Multiply 17 by 8 to get 136, and subtract this from 140 to get a remainder of 4.

Adding together what we have collected in the quotient (2, 0, 8), we get 208 as the quotient.

Therefore, the quotient is $\textbf{208}$ and the remainder is $\textbf{4}$.

The correct option from the choices is: $208$ with a remainder of 4.

Therefore, the solution to the problem is $208$ with a remainder of 4.

To solve this problem using long division, we will follow these steps:

• Step 1: Set up the division of 5111 by 11.
• Step 2: Divide the first few digits until you have a number greater than or equal to 11.
• Step 3: Subtract the result and bring down the next digit.
• Step 4: Repeat the process until all digits are used.

Let's perform the division:

Step 1: Take the first digit of 5111, which is 5. Since 5 is less than 11, look at the first two digits, 51.

Step 2: Divide 51 by 11. The result is 4 (since $4 \times 11 = 44$), with a remainder of 7.

Subtract 44 from 51, we get 7. Bring down the next digit, 1, making the number 71.

Step 3: Divide 71 by 11, which is 6 (since $6 \times 11 = 66$), with a remainder of 5.

Subtract 66 from 71, we get 5. Bring down the next digit, 1, making the number 51.

Step 4: Divide 51 by 11, which is 4 again, with a remainder of 7 (since $4 \times 11 = 44$). There are no more digits to bring down.

The quotient from dividing 5111 by 11 is therefore 464 with a remainder of 7.

Therefore, the solution to the problem is $464$ with a remainder of 7.

To solve the division of 7235 by 28, we will use the long division method:

• Step 1: Determine how many times 28 can go into the first part of the number, 72. Since $28 \times 2 = 56$ and $28 \times 3 = 84$, 28 fits into 72 twice. Write 2 on top.
• Step 2: Subtract $56$ from $72$ to get $16$. Bring down the next digit, which is 3, making it 163.
• Step 3: Determine how many times 28 fits into 163. Since $28 \times 5 = 140$ and $28 \times 6 = 168$, 28 fits in 163 five times. Write 5 on top next to 2.
• Step 4: Subtract $140$ from $163$ to get $23$. Bring down the final digit, 5, making it 235.
• Step 5: Determine how many times 28 fits into 235. Since $28 \times 8 = 224$ and $28 \times 9 = 252$, 28 fits into 235 eight times. Write 8 on top next to the 25, making our partial quotient 258.
• Step 6: Subtract $224$ from $235$ to get a remainder of $11$.

Therefore, when dividing 7235 by 28, the quotient is $258$ with a remainder of $11$.

The correct answer to the multiple-choice question is $\text{Option 2:}$ $258$ with a remainder of $11$.

To solve this problem, we'll perform a long division of 21350 by 20:

• Step 1: Divide 21 (the first two digits of 21350) by 20. This gives us a quotient of 1. Multiply 1 by 20 and subtract from 21 to get a remainder of 1.
• Step 2: Bring down the next digit (3) from the dividend, making it 13. The current number is now 13. Divide 13 by 20, which gives a quotient of 0. Bring down the next digit (5), making it 135.
• Step 3: Divide 135 by 20, which gives a quotient of 6. Multiply 6 by 20 to get 120, and subtract from 135 to have a remainder of 15.
• Step 4: Bring down the final digit (0), making it 150. Divide 150 by 20, which yields a quotient of 7. Multiply 7 by 20 to get 140. Subtract from 150 to leave a remainder of 10.

The entire division gives us a quotient of 1067 and a remainder of 10.

To verify: Multiply the quotient 1067 by 20 to get 21340, then add the remainder 10 to get back to the original dividend 21350.

Therefore, the solution to the problem is $1067$ with a remainder of 10.

To solve this problem, we will conduct the long division of 34202 by 23.

• Step 1: Set up the long division, writing 34202 under the division bar and 23 outside.
• Step 2: Start with the leftmost digits. Since 23 does not go into 3, consider the next digit, making it 34.
• Step 3: Divide 34 by 23 to get 1. Place the 1 above the division bar.
• Step 4: Multiply 1 by 23 to get 23, and subtract from 34 to get a remainder of 11.
• Step 5: Bring down the next digit, making it 110.
• Step 6: Divide 110 by 23 to get 4. Place the 4 above the division bar.
• Step 7: Multiply 4 by 23 to get 92, and subtract from 110 to get a remainder of 18.
• Step 8: Bring down the next digit, making it 182.
• Step 9: Divide 182 by 23 to get 7. Place the 7 above the division bar.
• Step 10: Multiply 7 by 23 to get 161, and subtract from 182 to get a remainder of 21.
• Step 11: Bring down the last digit, making it 210.
• Step 12: Divide 210 by 23 to get 9. Place the 9 above the division bar.
• Step 13: Multiply 9 by 23 to get 207, and subtract from 210 to get a remainder of 3.
• Step 14: The quotient is 1487, and there is a remainder of 1.

Through this process, we find that when 34202 is divided by 23, the quotient is $1487$ with a remainder of 1.

Therefore, the correct answer is $1487$ with a remainder of 1.

To solve this problem, we'll use long division to divide 20104 by 21:

• Start with the first two digits of the dividend, 20. Since 21 is greater than 20, we consider the first three digits, 201.

• Determine how many times 21 goes into 201. It fits 9 times (since $21 \times 9 = 189$).

• Subtract 189 from 201 to get 12.

• Bring down the next digit of the dividend (0) to make 120.

• Determine how many times 21 fits into 120. It fits 5 times (since $21 \times 5 = 105$).

• Subtract 105 from 120 to get 15.

• Bring down the last digit of the dividend (4) to make 154.

• Determine how many times 21 fits into 154. It fits 7 times (since $21 \times 7 = 147$).

• Subtract 147 from 154 to get 7, which is the remainder.

The quotient is 957 and the remainder is 7.

Therefore, the solution to the problem is $957$ with a remainder of 7.

To solve this problem, we will perform a long division of 52103 by 29 and determine both the quotient and the remainder.

Let's divide 52103 by 29 using long division:

• First, see how many times 29 can go into the first few digits of 52103, starting from the left.
• 29 into 52 goes 1 time. Multiply $29 \times 1 = 29$. Subtract 29 from 52, giving us 23.
• Bring down the next digit forming 231.
• 29 into 231 goes 7 times. Multiply $29 \times 7 = 203$. Subtract 203 from 231, giving us 28.
• Bring down the next digit forming 280.
• 29 into 280 goes 9 times. Multiply $29 \times 9 = 261$. Subtract 261 from 280, giving us 19.
• Bring down the last digit forming 193.
• 29 into 193 goes 6 times. Multiply $29 \times 6 = 174$. Subtract 174 from 193, giving us 19.

The quotient of the division of 52103 by 29 is 1796 with a remainder of 19, which matches choice 4.

Therefore, the correct answer is: $1796$ with a remainder of 19.