Long Division: Division of a 3-digit number without a remainder

To solve the problem of dividing 306 by 2 using long division, we will follow these steps:

• Step 1: Set up the expression as $306 \div 2$.
• Step 2: Begin the division with the hundreds digit.
• Step 3: Move to the tens digit and then finally to the units digit.

Let's proceed with these steps:

Step 1: Start with the hundreds place. Divide 3 (from 306) by 2. The quotient is 1, and we write it atop. Multiply 1 by 2, which gives 2, and subtract it from 3, leaving a remainder of 1. Bring down the next digit (0).

Step 2: Divide 10 by 2. The quotient is 5. Write 5 atop the division bar. Multiply 5 by 2, resulting in 10. Subtract 10 from 10, leaving no remainder. Bring down the next digit (6).

Step 3: Divide 6 by 2. The quotient is 3. Write 3 directly next to the previous quotient digits (1 and 5) on top. Multiply 3 by 2, which gives 6. Subtract this from 6, which yields 0, showing no remainder.

Thus, by performing long division, we find that:

The solution to the division $306 \div 2$ is $\mathbf{153}$.

To solve this problem, we'll perform long division of 216 by 3:

• Step 1: Divide the hundreds digit. 2 (from 216) divided by 3 is 0, so we write 0 as the quotient.
• Step 2: Since 2 is less than 3, we consider the next digit. Thus, we take the first two digits, 21. 21 divided by 3 equals 7.
• Step 3: Write 7 in the quotient beside the 0. The quotient now is 07.
• Step 4: Now, consider the last digit: Bring down the 6. Divide 16 by 3 to get 5. Write 6 in the remainder position.
• Step 5: Since 6 is divisible by 3 evenly, divide it to get 2. Thus, the final digit for our quotient is 2.
• Step 6: Combining these, the full quotient becomes 72.

Therefore, the solution to the division $\frac{216}{3}$ is $72$.

To solve this problem, we will perform long division of 603 by 3. Below are the steps explained in detail:

• Step 1: Start with the leftmost digit of the dividend (603). This is 6. Divide 6 by 3 to get 2. So, we place 2 above the line, indicating the first digit of the quotient.
• Step 2: Multiply 2 (the quotient digit we've obtained) by 3 (the divisor) to get 6. Subtract this 6 from the first digit of 603, which gives us 0.
• Step 3: Bring down the next digit of the dividend, which is 0. The result is now considered as 0.
• Step 4: Divide this 0 by 3, which gives 0 for the next digit in the quotient. Place this 0 alongside the 2 above the line.
• Step 5: Bring down the final digit of the dividend, which is 3.
• Step 6: Divide the 3 by 3 to get 1. Place 1 in the quotient line, forming the last digit.
• Step 7: Multiply 1 by 3 to get 3, subtract this 3 from the digit brought down (which is also 3), resulting in 0.

Thus, combining all digits in the quotient gives us 201.

Therefore, the solution to this problem is $201$.

To solve this problem, we'll perform long division of 208 by 4:

• Step 1: Begin by setting up the division. Write 208 (dividend) under the division bar and 4 (divisor) on the outside.
• Step 2: Determine how many times 4 can fit into the first digit of 208, which is 2. Since 4 does not go into 2, we consider the first two digits, 20.
• Step 3: 4 fits into 20 exactly 5 times (since 4 × 5 = 20). Write 5 above the division bar above the zero in 20.
• Step 4: Multiply 5 by 4 to get 20, and subtract 20 from 20, leaving a remainder of 0. Bring down the next digit, which is 8.
• Step 5: Now, determine how many times 4 goes into 8. It goes exactly 2 times (since 4 × 2 = 8). Write 2 above the division bar, next to 5.
• Step 6: Multiply 2 by 4 to get 8, subtract from 8 to leave a remainder of 0.

The division is complete, and we find that 208 divided by 4 equals $52$.

Therefore, the correct answer is $52$.

To solve the division of $100$ by $5$ using long division, we follow these steps:

• Step 1: Divide the leftmost digit of the dividend $1$ by the divisor $5$. Since $1$ is smaller than $5$, we look at the first two digits of the dividend $10$.
• Step 2: Divide $10$ by $5$ to get $2$. Write $2$ above the dividend.
• Step 3: Multiply $2$ by $5$, giving us $10$. Subtract $10$ from $10$ to get $0$.
• Step 4: Bring down the next digit $0$. Divide this $0$ by $5$, which gives us $0$. Write $0$ next to the $2$.
• Step 5: Multiply this $0$ by $5$, which also gives $0$. Subtract $0$ from $0$ to get $0$ again.

The quotient of this division is $20$. Therefore, the result of dividing $100$ by $5$ is $20$.

To solve this problem, we'll perform long division:

• **Step 1**: Write down the division as $133 \div 7$.
• **Step 2**: Start with the leading digit or first two digits, $13$. Determine how many times 7 can fit into 13. The number 7 fits into 13 a total of 1 time.
• **Step 3**: Multiply 1 by 7 to get 7, and subtract 7 from 13 to get a remainder of 6.
• **Step 4**: Bring down the next digit, which is 3, making the number 63.
• **Step 5**: Determine how many times 7 can fit into 63. The number 7 fits into 63 a total of 9 times.
• **Step 6**: Multiply 9 by 7 to get 63. Subtract 63 from 63 to obtain a remainder of 0.

Therefore, the division results in $19$ with no remainder.

The solution to the problem is $19$.

To solve this problem, we'll utilize the long division method to divide 252 by 7:

1. Start by dividing the first digit (2) of 252 by 7. Since 2 is less than 7, consider the first two digits (25).
2. Divide 25 by 7. The largest integer quotient is 3 because $3 \times 7 = 21$. Write 3 above the division bar.
3. Subtract 21 from 25 to get a remainder of 4.
4. Bring down the next digit (2) to make the new number 42.
5. Divide 42 by 7. The quotient is 6 because $6 \times 7 = 42$.
6. Subtract 42 from 42, resulting in a remainder of 0.

Since there are no more digits to bring down and the remainder is 0, the division is complete.

The quotient of 252 divided by 7 is $36$.

Comparing this result with the given choices, we see that choice 3 is correct.

To solve the problem of dividing 518 by 7, we'll perform long division:

1. Start with the first digit of the dividend, which is 5, in the hundreds place.

• $7$ does not go into $5$, so proceed to the next digit to make it 51.

2. Determine how many times $7$ goes into $51$ completely:

• $7 \times 7 = 49$, which is closest to 51 without exceeding it.
• Write $7$ as part of the quotient.
• Subtract $49$ from $51$, which leaves a remainder of $2$.

3. Bring down the next digit from the dividend, which is $8$, making the new number $28$.

4. Determine how many times $7$ goes into $28$ completely:

• $7 \times 4 = 28$, which equals 28.
• Write $4$ as the next digit of the quotient.
• Subtract $28$ from $28$, which leaves a remainder of $0$.

Since we reach a remainder of 0 after using all digits of the dividend, the division is exact. Therefore, the result is:

$74$. Thus, the correct answer is choice 3.

To determine the result of dividing 184 by 8, we use long division:

1. **Set up the division**: Write 184 under the division bracket with 8 outside.
2. **Analyze the first digit**: 8 goes into 18, the first two digits of 184, a total of 2 times since $8 \times 2 = 16$.
3. **Subtract 16 from 18**: We get a remainder of 2. Bring down the next digit 4, giving us 24.
4. **Divide again**: 8 goes into 24 exactly 3 times because $8 \times 3 = 24$.
5. **Subtract**: Subtracting gives zero remainder.

Thus, the quotient is 23, confirming that $184 \div 8 = 23$ without any remainder. The operation has been checked accurately by multiplying 23 by 8, which yields 184.

Therefore, the solution to the division problem is $23$. This matches the correct choice provided in the list of options.

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

• Step 1: Consider the dividend, 200, and the divisor, 8.
• Step 2: Use long division to find the quotient.
• Step 3: Perform the necessary calculations.

Now, let's work through each step:
Step 1: The dividend is 200, and the divisor is 8.
Step 2: We’ll perform the division using long division.
Step 3: Divide the first digit 2 by 8. It goes 0 times, so consider the first two digits, 20.
20 divided by 8 is 2 with a remainder of 4.
Write down 2 and bring down the next digit, making it 40.
40 divided by 8 is 5.
Thus, the complete division gives 25 without a remainder.

Therefore, the solution to the problem is $25$.

To solve this problem, we'll divide 640 by 8 using long division.

• Step 1: Set up the long division, where 640 is the dividend inside the division bracket, and 8 is the divisor outside.

• Step 2: Determine how many times 8 goes into the first digit, 6. It does not go into 6 since 6 is less than 8. Move to the next digit.

• Step 3: Consider the first two digits, 64. 8 goes into 64 a total of 8 times since $8 \times 8 = 64$. Write 8 above the division line.

• Step 4: Subtract 64 from 64, which results in 0. Bring down the next digit, 0, from the dividend, making it 00.

• Step 5: Determine how many times 8 goes into 0. It goes 0 times, so write 0 as the next digit of the quotient.

• Step 6: The final required division is complete. The quotient is $80$.

The result of dividing 640 by 8 is $80$. Therefore, the correct answer is choice 3: $80$.

We will use the long division method to solve for the quotient when dividing 171 by 9:

• Step 1: Set up the division: Place 171 inside the division bracket, and 9 outside. We will divide 171 by 9.
• Step 2: Divide the first digit: 9 goes into 17 once. Write 1 above the division line.
• Step 3: Multiply and subtract: Multiply 1 by 9 to get 9. Subtract 9 from 17 to get a remainder of 8.
• Step 4: Bring down the next digit: Bring down the next digit (1) from 171, resulting in 81.
• Step 5: Divide the new number: 9 goes into 81 nine times. Write 9 next to the 1 above the division line.
• Step 6: Multiply and subtract: Multiply 9 by 9 to get 81. Subtract 81 from 81, resulting in 0 as the final remainder.

The quotient of 171 divided by 9 is $\mathbf{19}$.

Therefore, the solution to the problem is $19$.