Answer: 4

Set up the division: Distributing 32 pencils equally among 8 students requires $32 \div 8$. Apply the multiplication fact: $8 \times 4 = 32$, so each student gets $4$ pencils. Why distractors fail: Option A ($3$): $8 \times 3 = 24 \neq 32$. Option C ($5$): $8 \times 5 = 40 \neq 32$. Option D ($6$): $8 \times 6 = 48 \neq 32$.

Answer: $18 \div 2 = 9$ and $18 \times 0.5 = 9$, so both operations yield the same result

Connect halving to dividing by 2: Dividing by 2 means splitting into 2 equal groups. Multiplying by $0.5$ (one-half) achieves the identical result. Both yield $9$ when applied to $18$. Why the correct answer works: Option A correctly demonstrates that $18 \div 2$ and $18 \times 0.5$ both produce $9$, showing the equivalence of halving and dividing by 2. Why distractors fail: Option B is coincidental — subtracting the quotient from the dividend is not the same as dividing by 2 in general (e.g., $10 \div 2 = 5$, but $10 - 5 = 5$ only works here by coincidence). Option C is wrong because a divisor being even does not guarantee it divides all numbers evenly — for example, $3 \div 2 = 1.5$, which is not a whole number. Option D is wrong because individual digit divisibility does not determine whether the whole number is divisible — for example, $13 \div 2 = 6.5$, even though the digit 1 is odd.

Answer: The answer is incorrect because $8 \times 6 = 48$, not $54$

Use the multiplication check: To verify a division result with no remainder, multiply the quotient by the divisor. The product must equal the original dividend exactly. Why the correct answer works: Option B is correct: $8 \times 6 = 48 \neq 54$, so the quotient of $8$ is wrong. The correct answer is $54 \div 6 = 9$ because $9 \times 6 = 54$. Why distractors fail: Option A is wrong because 'close' is not sufficient — the product must exactly equal the dividend. Option C incorrectly introduces a remainder when the problem divides evenly. Option D is wrong because multiplying quotient by divisor is a standard and reliable check.

Answer: Because $6 \times 8 = 48$

Division as the inverse of multiplication: Division asks: how many groups of the divisor fit into the dividend? The answer (quotient) satisfies $\text{divisor} \times \text{quotient} = \text{dividend}$. Why the correct answer works: Option C is correct because $6 \times 8 = 48$ directly shows that 6 fits into 48 exactly 8 times. Why distractors fail: Option A confuses subtraction with division. Option B uses addition, which does not explain division. Option D uses an irrelevant addition and subtraction sequence.

Answer: 6

Recall the multiplication fact: To solve $36 \div 6$, think: what number times 6 equals 36? Since $6 \times 6 = 36$, the answer is $6$. Why the correct answer works: Option B ($6$) is correct because $6 \times 6 = 36$, which matches the dividend. Why distractors fail: Option A ($5$) is wrong because $5 \times 6 = 30$. Option C ($7$) is wrong because $7 \times 6 = 42$. Option D ($8$) is wrong because $8 \times 6 = 48$.

Answer: No, because the correct check is $6 \times 5 = 30$, using multiplication

Identify the correct verification method: Division is the inverse of multiplication, so the check for $30 \div 5 = 6$ is $6 \times 5 = 30$. Why the correct answer works: Option C is correct: multiplying the quotient ($6$) by the divisor ($5$) should return the dividend ($30$), and indeed $6 \times 5 = 30$. Why distractors fail: Options A and B incorrectly accept addition as a valid check — addition is not the inverse of division. Option D suggests subtraction, which is also not the inverse of division.

Answer: Both methods are valid, though Student A's is more efficient

Analyze Student A's method: Student A uses the standard multiplication check: $8 \times 9 = 72$. This directly confirms the quotient is $9$. Analyze Student B's method: Student B adds $9$ eight times: $9 \times 8 = 72$. Repeated addition is equivalent to multiplication, so this is also valid, just less efficient. Why distractors fail: Options A and B each dismiss one valid method. Option D is wrong because subtraction is not the inverse of division — multiplication (or its equivalent, repeated addition) is.

Answer: 18

Find the original number: From $n \div 4 = 9$, we get $n = 9 \times 4 = 36$. Divide by 2: $36 \div 2 = 18$. Why distractors fail: Option A ($9$) is the quotient when dividing by 4, not by 2. Option B ($12$) might come from $36 \div 3$. Option D ($36$) is $n$ itself, not $n \div 2$.

Answer: $6, 18, 24, 36, 48$

Identify the requirement: Each dividend must be a multiple of 6 so that dividing by 6 leaves no remainder. Check each set: Option A: $20 \div 6 = 3$ R $2$ ✗. Option B: $6, 18, 24, 36, 48$ — all are multiples of 6 ✓. Option C: $15 \div 6 = 2$ R $3$ ✗ and $33 \div 6 = 5$ R $3$ ✗. Option D: $35 \div 6 = 5$ R $5$ ✗. Why distractors fail: Options A, C, and D each contain at least one number that is not a multiple of 6, which means that division would leave a remainder.

Answer: Only Claim 2 is correct

Evaluate each claim: Claim 1 gives a quotient of 5, but $5 \times 5 = 25 \neq 30$, so it is wrong despite the convoluted addition reasoning. Claim 2 gives a quotient of 6, and $5 \times 6 = 30$ ✓. Claim 3 gives a quotient of 7, but subtracting 5 five times from 30 gives $30 - 25 = 5$, which means only 5 groups were subtracted with 5 left over — that means the quotient should be 6, not 7. Why the correct answer works: Option B is correct because only Claim 2 uses the proper multiplication check and arrives at the right answer. Why distractors fail: Option A is wrong because Claim 1 arrives at $5$, but $5 \times 5 = 25 \neq 30$. Option C is wrong because Claim 3 miscounts the number of subtractions. Option D is wrong because Claim 1 is incorrect.

Answer: 8

Use the 5s pattern: Dividing by 5 uses the 5s multiplication table. $5 \times 8 = 40$, so $40 \div 5 = 8$. Why the correct answer works: Option B ($8$) is correct because $5 \times 8 = 40$. Why distractors fail: Option A ($6$) is wrong because $5 \times 6 = 30$. Option C ($9$) is wrong because $5 \times 9 = 45$. Option D ($7$) is wrong because $5 \times 7 = 35$.

Answer: 42

Halve the number: Dividing by 2 means finding half. Half of $84$ is $42$ because $2 \times 42 = 84$. Why the correct answer works: Option B ($42$) is correct: $2 \times 42 = 84$. Why distractors fail: Option A ($41$) gives $2 \times 41 = 82$, not $84$. Option C ($44$) gives $2 \times 44 = 88$. Option D ($82$) subtracts 2 instead of dividing.

Answer: 35

Reverse the division using multiplication: If the original number $\div 5 = 7$, then the original number $= 7 \times 5 = 35$. Why the correct answer works: Option C ($35$) is correct because $35 \div 5 = 7$ with no remainder. Why distractors fail: Option A ($12$): $12 \div 5 = 2$ remainder $2$. Option B ($25$): $25 \div 5 = 5$, not $7$. Option D ($57$): $57 \div 5 = 11$ remainder $2$.

Answer: $9 \times 9 = 81$

Link division to multiplication: To solve $81 \div 9$, find the number that, when multiplied by 9, gives 81. Why the correct answer works: Option B ($9 \times 9 = 81$) is correct because it shows that $9$ groups of $9$ equal $81$, so $81 \div 9 = 9$. Why distractors fail: Option A ($9 \times 8 = 72$) gives the wrong dividend. Option C ($9 \times 10 = 90$) overshoots. Option D multiplies in the wrong direction — it does not help find the quotient.

Answer: 9

Recall the 8 times table: $8 \times 9 = 72$, so $72 \div 8 = 9$. Why the correct answer works: Option A ($9$) is correct because $8 \times 9 = 72$. Why distractors fail: Option B ($8$) is wrong because $8 \times 8 = 64$. Option C ($7$) is wrong because $8 \times 7 = 56$. Option D ($10$) is wrong because $8 \times 10 = 80$.

Answer: 9

Use the 7 times table: $7 \times 9 = 63$, so $63 \div 7 = 9$. Why the correct answer works: Option C ($9$) is correct because $7 \times 9 = 63$. Why distractors fail: Option A ($7$) is wrong because $7 \times 7 = 49$. Option B ($8$) is wrong because $7 \times 8 = 56$. Option D ($10$) is wrong because $7 \times 10 = 70$.

Answer: $42 \div 8$

Test each option: $24 \div 6 = 4$, $35 \div 7 = 5$, $42 \div 8 = 5.25$, $54 \div 9 = 6$. Only $42 \div 8$ does not divide evenly. Why the correct answer works: Option C ($42 \div 8$) is correct because $8 \times 5 = 40$ and $42 - 40 = 2$, leaving a remainder of 2. Why distractors fail: Options A, B, and D all divide evenly: $6 \times 4 = 24$, $7 \times 5 = 35$, and $9 \times 6 = 54$.

Answer: 56

Apply the rule for dividing by 1: Any number divided by $1$ equals itself. So $56 \div 1 = 56$. Why the correct answer works: Option D ($56$) is correct because dividing by $1$ always returns the original number. Why distractors fail: Option A ($0$) confuses dividing by 1 with multiplying by 0. Option B ($1$) confuses the divisor with the quotient. Option C ($55$) incorrectly subtracts 1 instead of dividing.

Answer: 24

Identify the requirement: The number must divide evenly by both 3 and 4, meaning it must be a multiple of both. Check each option: $18 \div 3 = 6$ ✓ but $18 \div 4 = 4.5$ ✗. $24 \div 3 = 8$ ✓ and $24 \div 4 = 6$ ✓. $25 \div 3 \approx 8.3$ ✗. $32 \div 3 \approx 10.7$ ✗. Why distractors fail: Option A ($18$) is divisible by 3 but not by 4. Option C ($25$) is divisible by neither 3 nor 4. Option D ($32$) is divisible by 4 but not by 3.

Answer: 5

Use the times table for 9: Ask: $9 \times ? = 45$. Since $9 \times 5 = 45$, the quotient is $5$. Why the correct answer works: Option C ($5$) is correct because $9 \times 5 = 45$. Why distractors fail: Option A ($4$) is wrong because $9 \times 4 = 36$. Option B ($6$) is wrong because $9 \times 6 = 54$. Option D ($7$) is wrong because $9 \times 7 = 63$.

Answer: 12

Set up the division: $60 \div 5 = ?$. Using the 5s pattern: $5 \times 12 = 60$. Why the correct answer works: Option C ($12$) is correct because $5 \times 12 = 60$. Why distractors fail: Option A ($10$): $5 \times 10 = 50$. Option B ($11$): $5 \times 11 = 55$. Option D ($15$): $5 \times 15 = 75$.

Answer: Yes, for any positive dividend, a smaller positive divisor always produces a larger quotient

Understand the relationship: For a fixed positive dividend, the quotient $= \frac{\text{dividend}}{\text{divisor}}$. As the divisor decreases (while remaining positive), the quotient increases. Verify with the example: $90 \div 5 = 18$ and $90 \div 2 = 45$. Since $2 < 5$, the quotient $45 > 18$, which confirms the principle. Why distractors fail: Option A is wrong — division follows a clear inverse relationship. Option B incorrectly restricts the rule to even dividends. Option D incorrectly restricts the rule to exact division; the relationship holds regardless of remainders.

Answer: 7

Set up the division: $49 \div 7 = ?$. From the 7 times table: $7 \times 7 = 49$. Why the correct answer works: Option B ($7$) is correct because $7 \times 7 = 49$, confirming 7 complete rows. Why distractors fail: Option A ($6$): $7 \times 6 = 42$. Option C ($8$): $7 \times 8 = 56$. Option D ($9$): $7 \times 9 = 63$.

Answer: "A gardener plants 56 seeds equally in 7 rows. How many seeds per row?" — Check: $8 \times 7 = 56$

Identify what makes a valid division word problem: The problem must require dividing 56 by 7 to find the answer, and the verification should multiply the quotient by the divisor to recover the dividend. Why the correct answer works: Option B sets up an equal-sharing scenario: $56 \div 7 = 8$ seeds per row. The check $8 \times 7 = 56$ correctly uses multiplication to verify. Why distractors fail: Option A asks how many days until 49 apples remain, which only requires one subtraction ($56 - 7 = 49$), not division. It does not require computing $56 \div 7$ at all. Option C is a subtraction problem ($56 - 7 = 49$ items sold), and the check shown ($49 \times 7 = 343$) is irrelevant and incorrect. Option D introduces a unit conversion (km to metres), which changes the problem to $56{,}000 \div 7$, not $56 \div 7$ as required.