Answer: Correct; $903 \times 5 = 4{,}515$

Verify the student's work: $903 \times 5 = 4{,}515$. ✓ The answer is correct. Trace the division: $45 \div 5 = 9$. $1 \div 5 = 0$ R$1$. Bring down $5$: $15 \div 5 = 3$. Quotient: $903$. Why distractors fail: Option B gives a five-digit number, which is far too large. Option C ($93$) drops the zero. Option D ($913$) incorrectly assigns $1$ to the tens place.

Answer: 625

Step-by-step long division: $56 \div 9 = 6$ remainder $2$. Bring down $2$: $22 \div 9 = 2$ remainder $4$. Bring down $5$: $45 \div 9 = 5$ remainder $0$. Why the correct answer works: The quotient is $625$. Verification: $625 \times 9 = 5{,}625$. Why distractors fail: Option B ($635$) uses an incorrect middle digit. Option C ($615$) confuses $22 \div 9$. Option D ($725$) miscalculates the first step.

Answer: $9{,}027$

Determine the required quotient form: The quotient must be a 4-digit number of the form $X{,}00Y$ (zeros in both tens and hundreds places). Check each option: $3{,}315 \div 3 = 1{,}105$ — hundreds digit is $1$, not $0$. $9{,}027 \div 3 = 3{,}009$ — hundreds digit is $0$, tens digit is $0$. ✓ $3{,}126 \div 3 = 1{,}042$ — hundreds is $0$ but tens is $4$, so only one zero. $6{,}321 \div 3 = 2{,}107$ — hundreds digit is $1$, not $0$. Why the correct answer works: $9{,}027 \div 3 = 3{,}009$, which has zeros in both the tens and hundreds places. Verification: $3{,}009 \times 3 = 9{,}027$. ✓ Why distractors fail: Option A gives $1{,}105$ — no zero in the hundreds place. Option C gives $1{,}042$ — has a zero in the hundreds place but not the tens place. Option D gives $2{,}107$ — no zero in the hundreds place.

Answer: Divide $6$ into $8$, because $8 > 6$

Deciding where to start: Compare the divisor to the first digit of the dividend. Since $8 \geq 6$, we can begin dividing into the first digit. Why the correct answer works: Because $8$ is greater than or equal to $6$, the divisor fits at least once, so we start there. This means the quotient will have three digits. Why distractors fail: Option B is wrong because you do not always start with two digits — only when the first digit is smaller than the divisor. Option C is wrong because long division proceeds digit by digit. Option D is wrong because you never skip the leading digit.

Answer: $5{,}637$

Use the division relationship: The relationship is: Dividend $= Q \times D + R = 805 \times 7 + 2$. Compute: $805 \times 7 = 5{,}635$. Adding the remainder: $5{,}635 + 2 = 5{,}637$. Why distractors fail: Option B ($5{,}635$) forgets to add the remainder. Option C ($5{,}639$) adds $4$ instead of $2$. Option D ($5{,}640$) adds $5$ instead of $2$.

Answer: $1{,}053$

Compute the correct quotient: $6 \div 6 = 1$. $3 \div 6 = 0$ R$3$. Bring down $1$: $31 \div 6 = 5$ R$1$. Bring down $8$: $18 \div 6 = 3$. Quotient: $1{,}053$. Identify the student's error: The student wrote $153$, which omits the zero in the hundreds place. This is a classic error of skipping the zero when the divisor doesn't fit into a partial dividend. Why distractors fail: Option A ($1{,}043$) has an incorrect tens digit. Option C ($1{,}530$) rearranges the digits. Option D ($1{,}503$) places the $0$ in the wrong position.

Answer: 408

Set up the division: We need $2{,}856 \div 7$. Perform the long division: $28 \div 7 = 4$. $5 \div 7 = 0$ remainder $5$. Bring down $6$: $56 \div 7 = 8$. Quotient is $408$. Verify: $408 \times 7 = 2{,}856$. ✓ Why distractors fail: Option A ($48$) omits the zero, a common error. Option C ($480$) places the zero incorrectly. Option D ($418$) has an error in the tens digit.

Answer: Quotient B has more digits

Analyze problem A: For $5{,}432 \div 8$: the first digit $5 < 8$, so we start with $54$. The quotient will have $4 - 1 = 3$ digits. Analyze problem B: For $5{,}432 \div 4$: the first digit $5 \geq 4$, so we start dividing immediately. The quotient will have $4$ digits. Why the correct answer works: Quotient B has 4 digits while Quotient A has 3 digits. Indeed: $5{,}432 \div 8 = 679$ and $5{,}432 \div 4 = 1{,}358$. Why distractors fail: Option A reverses the conclusion. Option C is wrong because the divisors differ in how they relate to the leading digit. Option D is wrong because we can predict quotient length from the divisor-to-leading-digit comparison.

Answer: $4{,}674$

Compute each quotient: $6{,}072 \div 6 = 1{,}012$ (contains zeros). $4{,}218 \div 6 = 703$ (contains a zero). $5{,}418 \div 6 = 903$ (contains a zero). $4{,}674 \div 6 = 779$ (no zeros). Why the correct answer works: $4{,}674 \div 6 = 779$ and $779$ contains no zero digits. Verification: $779 \times 6 = 4{,}674$. ✓ Why distractors fail: Options A, B, and C all produce quotients containing at least one zero digit.

Answer: The second DMSB cycle

Trace the division: Cycle 1: $72 \div 9 = 8$ R$0$. Cycle 2: Bring down $0$: $0 \div 9 = 0$ R$0$. Cycle 3: Bring down $3$: $3 \div 9 = 0$ R$3$. Quotient: $800$ R$3$. Why the correct answer works: The first zero appears in the second cycle when we try to divide $0$ by $9$. Why distractors fail: Option A is wrong — the first cycle gives $8$, not $0$. Option C is wrong — a zero appears even earlier, in cycle 2. Option D is wrong — the quotient $800$ contains two zeros.

Answer: The student forgot to place a zero in the quotient

Find the correct quotient: $4{,}032 \div 8 = 504$. The student wrote $54$, which omits the zero in the tens place. Why the correct answer works: After dividing $40 \div 8 = 5$, the next step is $3 \div 8 = 0$ remainder $3$. The student skipped writing the $0$ and jumped to the final digit. Why distractors fail: Option A is implausible since $54$ isn't the product of $4{,}032$ and $8$. Option C is wrong because the divisor $8$ yields the right answer when used correctly. Option D is partially related but the specific error is omitting the zero.

Answer: $1{,}013$ bags, $0$ kg left over

Perform the division: $8{,}104 \div 8$: $8 \div 8 = 1$. $1 \div 8 = 0$ R$1$. Bring down $0$: $10 \div 8 = 1$ R$2$. Bring down $4$: $24 \div 8 = 3$. Quotient: $1{,}013$, remainder $0$. Verify: $1{,}013 \times 8 = 8{,}104$. ✓ Why distractors fail: Option B ($1{,}130$) rearranges digits. Option C ($113$) drops a digit. Option D claims a remainder of $8$, but a remainder equal to the divisor means one more full bag should be counted.

Answer: 249

Perform the division step by step: Divide $1{,}245 \div 5$. Start: $12 \div 5 = 2$ remainder $2$. Bring down $4$ to get $24$. $24 \div 5 = 4$ remainder $4$. Bring down $5$ to get $45$. $45 \div 5 = 9$ remainder $0$. Why the correct answer works: The quotient is $249$ because $249 \times 5 = 1{,}245$. Why distractors fail: Option A ($239$) results from incorrectly computing the second digit. Option C ($259$) and Option D ($251$) arise from arithmetic errors in the middle or final steps.

Answer: 3

Determine quotient length: Since $7 < 8$, we start with the first two digits $70$. The quotient will have 3 digits, so there are 3 DMSB cycles (though the last cycle does not bring down a new digit). Why the correct answer works: $70 \div 8 = 8$ R$6$ → bring down $5$ → $65 \div 8 = 8$ R$1$ → bring down $6$ → $16 \div 8 = 2$ R$0$. That's 3 cycles yielding $882$. Why distractors fail: Option A (2) undercounts. Option C (4) would apply if the quotient had 4 digits. Option D (5) exceeds the number of digits in the dividend.

Answer: $4{,}218 \div 6$

Check each division: $1{,}248 \div 4 = 312$ (no zero). $1{,}575 \div 5 = 315$ (no zero). $4{,}218 \div 6 = 703$ (contains a zero). $3{,}456 \div 8 = 432$ (no zero). Why the correct answer works: $4{,}218 \div 6$: $42 \div 6 = 7$, $1 \div 6 = 0$ R$1$, bring down $8$: $18 \div 6 = 3$. Quotient: $703$, which has a zero in the tens place. Why distractors fail: Option A gives $312$, Option B gives $315$, and Option D gives $432$ — none contain a zero in the quotient.

Answer: 1,204

Divide each digit position: $3 \div 3 = 1$. $6 \div 3 = 2$. $0$ is obtained because $1 \div 3 = 0$ remainder $1$. Bring down $2$ to get $12$. $12 \div 3 = 4$. Why the correct answer works: The quotient is $1{,}204$ because $1{,}204 \times 3 = 3{,}612$. Note the zero in the tens place. Why distractors fail: Option B ($1{,}240$) places the zero in the wrong position. Option C ($1{,}024$) reverses some digits. Option D ($1{,}214$) omits the zero entirely.

Answer: $4{,}836$

Determine what the quotient must look like: We need a 3-digit quotient of the form $X0Y$ (zero in the tens place). Also, the first digit of the dividend must be less than $6$ for a 3-digit quotient to result. Check each option: $3{,}612 \div 6 = 602$ — zero in the hundreds place, not the tens place. $4{,}836 \div 6 = 806$ — zero in the tens place. ✓ $4{,}872 \div 6 = 812$ — no zero in the quotient at all. $3{,}456 \div 6 = 576$ — no zero in the quotient at all. Why the correct answer works: $4{,}836 \div 6$: $48 \div 6 = 8$, $3 \div 6 = 0$ remainder $3$, bring down $6$: $36 \div 6 = 6$. Quotient: $806$, which has a zero in the tens place. Verification: $806 \times 6 = 4{,}836$. ✓ Why distractors fail: Option A ($3{,}612 \div 6 = 602$) has the zero in the hundreds place, not the tens place. Option C ($4{,}872 \div 6 = 812$) produces no zero in the quotient. Option D ($3{,}456 \div 6 = 576$) also produces no zero in the quotient.

Answer: No, the quotient can have 3 digits if the leading digit of the dividend is smaller than the divisor

Evaluate the claim: Consider $2{,}400 \div 8 = 300$ (3 digits) versus $8{,}000 \div 2 = 4{,}000$ (4 digits). The number of quotient digits depends on whether the divisor fits into the first digit. Why the correct answer works: If the first digit is less than the divisor, we must combine it with the second digit, resulting in a quotient with one fewer digit. So the claim is false. Why distractors fail: Option A incorrectly affirms the claim. Option C goes too far in the other direction — the quotient can have 4 digits (e.g., $8{,}000 \div 2$). Option D is factually wrong; the divisor can be larger than some digits.

Answer: Student A is correct; Student B incorrectly placed a zero instead of computing $5$ in the tens place

Find the correct answer: $6{,}048 \div 8$: $60 \div 8 = 7$ R$4$. Bring down $4$: $44 \div 8 = 5$ R$4$. Bring down $8$: $48 \div 8 = 6$. Quotient: $756$. Verify: $756 \times 8 = 6{,}048$. ✓ Student A is correct. Analyze Student B's error: Student B wrote $706$, placing a $0$ in the tens place instead of $5$. This suggests Student B incorrectly judged that $8$ does not fit into $44$, or made a subtraction error yielding a partial dividend of less than $8$. Why other options fail: Option A's explanation is backwards. Option B incorrectly validates Student B. Option D gives an answer ($750$) that doesn't verify: $750 \times 8 = 6{,}000 \neq 6{,}048$.

Answer: 1,449

Set up the problem: $7{,}245 \div 5$. Perform the division: $7 \div 5 = 1$ R$2$. Bring down $2$: $22 \div 5 = 4$ R$2$. Bring down $4$: $24 \div 5 = 4$ R$4$. Bring down $5$: $45 \div 5 = 9$. Quotient: $1{,}449$. Why distractors fail: Option B ($1{,}459$) has a $5$ instead of a $4$ in the tens place. Option C ($1{,}349$) miscalculates the hundreds digit. Option D ($1{,}549$) overestimates the hundreds digit.

Answer: Tens place

Perform the division: $2{,}016 \div 4$: $20 \div 4 = 5$, $1 \div 4 = 0$ remainder $1$, bring down $6$ to get $16$, $16 \div 4 = 4$. The quotient is $504$. Why the correct answer works: In the quotient $504$: $5$ is in the hundreds place, $0$ is in the tens place, and $4$ is in the ones place. The zero is in the tens place. Why distractors fail: Option A (hundreds place) is wrong — the hundreds digit is $5$. Option C (ones place) is wrong — the ones digit is $4$. Option D (thousands place) is wrong — the quotient is only three digits.

Answer: Yes, the computation is correct

Verify the claim: $2{,}009 \times 4 = 8{,}036$. ✓ The student's answer is correct. Trace the division: $8 \div 4 = 2$. $0 \div 4 = 0$. $3 \div 4 = 0$ R$3$. Bring down $6$: $36 \div 4 = 9$. Quotient: $2{,}009$. Why distractors fail: Option B ($2{,}090$) places digits incorrectly. Option C ($2{,}019$) has the wrong tens digit. Option D ($209$) drops one of the zeros.

Answer: The claim is incorrect; $1{,}026 \div 3 = 342$, and skipping the zero changes the place values

Compute the correct answer: $1{,}026 \div 3$: $10 \div 3 = 3$ R$1$. Bring down $2$: $12 \div 3 = 4$. Bring down $6$: $6 \div 3 = 2$. Quotient: $342$. Why the claim is wrong: Every digit, including zero, holds a place value. Skipping the zero collapses $1{,}026$ into $126$, a completely different number. The correct quotient $342$ is very different from the claimed $42$. Why distractors fail: Option A and D both validate the incorrect claim. Option C gives $302$, which doesn't equal $1{,}026 \div 3$: $302 \times 3 = 906 \neq 1{,}026$.

Answer: Because after subtracting, the current partial dividend is smaller than the divisor, so $0$ is recorded and the next digit is brought down

Trace the division: $30 \div 6 = 5$, remainder $0$. Bring down $2$: $2 \div 6 = 0$ remainder $2$. Bring down $4$: $24 \div 6 = 4$. Quotient: $504$. Why the correct answer works: The zero appears because the partial dividend $2$ is less than the divisor $6$. We must write $0$ in that quotient position before bringing down the next digit. Why distractors fail: Option A is false — $6$ does not divide evenly into every digit. Option B is a common misconception; a $0$ in the dividend does not automatically cause a $0$ in the quotient. Option D is an invented rule with no basis.

Answer: 3 digits

Estimate quotient length: Compare $7$ to the first digit $4$. Since $4 < 7$, the divisor doesn't fit into the first digit alone, so we look at the first two digits: $43$. Since $43 \geq 7$, we start dividing there. The quotient will have one fewer digit than the dividend. Why the correct answer works: The dividend has 4 digits and we start from the second position, so the quotient has $4 - 1 = 3$ digits. Indeed $4{,}368 \div 7 = 624$. Why distractors fail: Option A (2 digits) underestimates. Option C (4 digits) would require the first digit to be ≥ 7. Option D (5 digits) is impossible since the quotient cannot exceed the dividend in length when dividing by a number > 1.

Answer: $406$ packages per route, $0$ left over

Perform the division: $36 \div 9 = 4$. $5 \div 9 = 0$ R$5$. Bring down $4$: $54 \div 9 = 6$. Quotient: $406$, remainder $0$. Verify: $406 \times 9 = 3{,}654$. ✓ Why distractors fail: Option A ($405$ R$9$) is incorrect; a remainder of $9$ equals the divisor, meaning another unit should be added. Option C ($46$) omits the zero. Option D ($400$ R$54$) stops the division too early.

Answer: Quotient $2{,}318$, remainder $3$

Perform the long division: $9 \div 4 = 2$ R$1$. Bring down $2$: $12 \div 4 = 3$ R$0$. Bring down $7$: $7 \div 4 = 1$ R$3$. Bring down $5$: $35 \div 4 = 8$ R$3$. State the result: Quotient: $2{,}318$, Remainder: $3$. Check: $2{,}318 \times 4 + 3 = 9{,}272 + 3 = 9{,}275$. ✓ Why distractors fail: Option B has the wrong remainder. Option C rounds up the quotient incorrectly. Option D has a zero where a $1$ should be.

Answer: 273

Perform the division: $16 \div 6 = 2$ remainder $4$. Bring down $3$: $43 \div 6 = 7$ remainder $1$. Bring down $8$: $18 \div 6 = 3$. Quotient: $273$. Verify: $273 \times 6 = 1{,}638$. ✓ Why distractors fail: Option B ($263$) miscalculates the tens digit. Option C ($278$) and Option D ($283$) arise from incorrect subtraction steps.

Answer: 3 digits

Compare divisor to leading digit: The first digit of $978$ is $9$. Since $9 \geq 9$, the divisor fits into the first digit, so the quotient will have the same number of digits as the dividend. Why the correct answer works: The dividend has 3 digits and the divisor fits into the first digit, so the quotient has 3 digits. Indeed, $978 \div 9 = 108$. Why distractors fail: Option A and Option B underestimate the length. Option D overestimates — a 3-digit number divided by a single digit cannot produce 4 digits.

Answer: 8

Decide where to start: Since $6 < 8$, we consider the first two digits: $64$. Now divide $64 \div 8 = 8$. Why the correct answer works: The first digit of the quotient is $8$ because $8 \times 8 = 64$ exactly. Why distractors fail: Option A ($7$) underestimates: $7 \times 8 = 56$, leaving too large a remainder. Option C ($6$) confuses the first digit of the dividend with the first quotient digit. Option D ($9$) overestimates: $9 \times 8 = 72 > 64$.