← 3-2 · Recover the dividend from quotient and remainder · Divisibility and Remainder Reasoning

Recover the dividend from quotient and remainder · 12 practice problems

3.OA.B.63.OA.C.7

Generated variants — 12

Freshly produced from the archetype’s parameters — problem, figure, and solution derived together.

Variant 1 answer: Quotient 11, remainder 0

When a certain number is divided by $6$ , the quotient is $7$ and the remainder is $2$ . Find the quotient and the remainder when this number is divided by $4$ .

Show solution

Understand

A number divided by 6 gives quotient 7 and remainder 2. We first recover that number, then divide it by 4 to find the new quotient and remainder.

Givens

The number divided by 6 has quotient 7 and remainder 2.
We then divide the same number by 4.

Unknowns

The quotient and remainder when the number is divided by 4.

Constraints

Remainders must be less than the divisor (less than 6 here, less than 4 in the second division).
The number is a whole number.

Plan

#11 Work Backwards · also uses: #6 Guess and Check

Rebuild the original number from quotient times divisor plus remainder, then carry out the new division by 4.

Execute

#11 Work Backwards 3.OA.B.6

The number equals 6 times the quotient 7, plus the remainder 2.

6 \times 7 + 2 = 42 + 2 = 44

Multiplying the quotient by the divisor and adding the remainder undoes the division.

#6 Guess and Check 3.OA.C.7

Now divide 44 by 4. 4 goes into 44 11 times, using 44, and 0 are left over.

44 \div 4 = 11 \cdots 0

Since 4 times 11 is 44 and 44 minus 44 is 0, the quotient is 11 and the remainder 0 (less than 4).

Answer: Quotient 11, remainder 0

Review

Check: 4 times 11 plus 0 equals 44, the recovered number, and the remainder 0 is less than the divisor 4, so the answer is valid.

Convert to an equation (tool 13): the number N satisfies N = 6 times 7 + 2 = 44, then solve N = 4q + r with 0 less than or equal to r less than 4, giving q = 11, r = 0.

Standards · min grade 3

3.OA.B.6 Understand division as an unknown-factor problem — Rebuilding the number 44 from quotient 7 and remainder 2.
3.OA.C.7 Fluently multiply and divide within 100 — Dividing 44 by 4 to get quotient 11 and remainder 0.

💡 This only needs Grade 3 division: multiply back to find the number, then divide it again!

Variant 2 answer: Quotient 8, remainder 1

When a certain number is divided by $5$ , the quotient is $9$ and the remainder is $4$ . Find the quotient and the remainder when this number is divided by $6$ .

Show solution

Understand

A number divided by 5 gives quotient 9 and remainder 4. We first recover that number, then divide it by 6 to find the new quotient and remainder.

Givens

The number divided by 5 has quotient 9 and remainder 4.
We then divide the same number by 6.

Unknowns

The quotient and remainder when the number is divided by 6.

Constraints

Remainders must be less than the divisor (less than 5 here, less than 6 in the second division).
The number is a whole number.

Plan

#11 Work Backwards · also uses: #6 Guess and Check

Rebuild the original number from quotient times divisor plus remainder, then carry out the new division by 6.

Execute

#11 Work Backwards 3.OA.B.6

The number equals 5 times the quotient 9, plus the remainder 4.

5 \times 9 + 4 = 45 + 4 = 49

Multiplying the quotient by the divisor and adding the remainder undoes the division.

#6 Guess and Check 3.OA.C.7

Now divide 49 by 6. 6 goes into 49 8 times, using 48, and 1 are left over.

49 \div 6 = 8 \cdots 1

Since 6 times 8 is 48 and 49 minus 48 is 1, the quotient is 8 and the remainder 1 (less than 6).

Answer: Quotient 8, remainder 1

Review

Check: 6 times 8 plus 1 equals 49, the recovered number, and the remainder 1 is less than the divisor 6, so the answer is valid.

Convert to an equation (tool 13): the number N satisfies N = 5 times 9 + 4 = 49, then solve N = 6q + r with 0 less than or equal to r less than 6, giving q = 8, r = 1.

Standards · min grade 3

3.OA.B.6 Understand division as an unknown-factor problem — Rebuilding the number 49 from quotient 9 and remainder 4.
3.OA.C.7 Fluently multiply and divide within 100 — Dividing 49 by 6 to get quotient 8 and remainder 1.

💡 This only needs Grade 3 division: multiply back to find the number, then divide it again!

Variant 3 answer: Quotient 7, remainder 1

When a certain number is divided by $5$ , the quotient is $8$ and the remainder is $3$ . Find the quotient and the remainder when this number is divided by $6$ .

Show solution

Understand

A number divided by 5 gives quotient 8 and remainder 3. We first recover that number, then divide it by 6 to find the new quotient and remainder.

Givens

The number divided by 5 has quotient 8 and remainder 3.
We then divide the same number by 6.

Unknowns

The quotient and remainder when the number is divided by 6.

Constraints

Remainders must be less than the divisor (less than 5 here, less than 6 in the second division).
The number is a whole number.

Plan

#11 Work Backwards · also uses: #6 Guess and Check

Rebuild the original number from quotient times divisor plus remainder, then carry out the new division by 6.

Execute

#11 Work Backwards 3.OA.B.6

The number equals 5 times the quotient 8, plus the remainder 3.

5 \times 8 + 3 = 40 + 3 = 43

Multiplying the quotient by the divisor and adding the remainder undoes the division.

#6 Guess and Check 3.OA.C.7

Now divide 43 by 6. 6 goes into 43 7 times, using 42, and 1 are left over.

43 \div 6 = 7 \cdots 1

Since 6 times 7 is 42 and 43 minus 42 is 1, the quotient is 7 and the remainder 1 (less than 6).

Answer: Quotient 7, remainder 1

Review

Check: 6 times 7 plus 1 equals 43, the recovered number, and the remainder 1 is less than the divisor 6, so the answer is valid.

Convert to an equation (tool 13): the number N satisfies N = 5 times 8 + 3 = 43, then solve N = 6q + r with 0 less than or equal to r less than 6, giving q = 7, r = 1.

Standards · min grade 3

3.OA.B.6 Understand division as an unknown-factor problem — Rebuilding the number 43 from quotient 8 and remainder 3.
3.OA.C.7 Fluently multiply and divide within 100 — Dividing 43 by 6 to get quotient 7 and remainder 1.

💡 This only needs Grade 3 division: multiply back to find the number, then divide it again!

Variant 4 answer: Quotient 11, remainder 3

When a certain number is divided by $7$ , the quotient is $6$ and the remainder is $5$ . Find the quotient and the remainder when this number is divided by $4$ .

Show solution

Understand

A number divided by 7 gives quotient 6 and remainder 5. We first recover that number, then divide it by 4 to find the new quotient and remainder.

Givens

The number divided by 7 has quotient 6 and remainder 5.
We then divide the same number by 4.

Unknowns

The quotient and remainder when the number is divided by 4.

Constraints

Remainders must be less than the divisor (less than 7 here, less than 4 in the second division).
The number is a whole number.

Plan

#11 Work Backwards · also uses: #6 Guess and Check

Rebuild the original number from quotient times divisor plus remainder, then carry out the new division by 4.

Execute

#11 Work Backwards 3.OA.B.6

The number equals 7 times the quotient 6, plus the remainder 5.

7 \times 6 + 5 = 42 + 5 = 47

Multiplying the quotient by the divisor and adding the remainder undoes the division.

#6 Guess and Check 3.OA.C.7

Now divide 47 by 4. 4 goes into 47 11 times, using 44, and 3 are left over.

47 \div 4 = 11 \cdots 3

Since 4 times 11 is 44 and 47 minus 44 is 3, the quotient is 11 and the remainder 3 (less than 4).

Answer: Quotient 11, remainder 3

Review

Check: 4 times 11 plus 3 equals 47, the recovered number, and the remainder 3 is less than the divisor 4, so the answer is valid.

Convert to an equation (tool 13): the number N satisfies N = 7 times 6 + 5 = 47, then solve N = 4q + r with 0 less than or equal to r less than 4, giving q = 11, r = 3.

Standards · min grade 3

3.OA.B.6 Understand division as an unknown-factor problem — Rebuilding the number 47 from quotient 6 and remainder 5.
3.OA.C.7 Fluently multiply and divide within 100 — Dividing 47 by 4 to get quotient 11 and remainder 3.

💡 This only needs Grade 3 division: multiply back to find the number, then divide it again!

Variant 5 answer: Quotient 7, remainder 0

When a certain number is divided by $6$ , the quotient is $8$ and the remainder is $1$ . Find the quotient and the remainder when this number is divided by $7$ .

Show solution

Understand

A number divided by 6 gives quotient 8 and remainder 1. We first recover that number, then divide it by 7 to find the new quotient and remainder.

Givens

The number divided by 6 has quotient 8 and remainder 1.
We then divide the same number by 7.

Unknowns

The quotient and remainder when the number is divided by 7.

Constraints

Remainders must be less than the divisor (less than 6 here, less than 7 in the second division).
The number is a whole number.

Plan

#11 Work Backwards · also uses: #6 Guess and Check

Rebuild the original number from quotient times divisor plus remainder, then carry out the new division by 7.

Execute

#11 Work Backwards 3.OA.B.6

The number equals 6 times the quotient 8, plus the remainder 1.

6 \times 8 + 1 = 48 + 1 = 49

Multiplying the quotient by the divisor and adding the remainder undoes the division.

#6 Guess and Check 3.OA.C.7

Now divide 49 by 7. 7 goes into 49 7 times, using 49, and 0 are left over.

49 \div 7 = 7 \cdots 0

Since 7 times 7 is 49 and 49 minus 49 is 0, the quotient is 7 and the remainder 0 (less than 7).

Answer: Quotient 7, remainder 0

Review

Check: 7 times 7 plus 0 equals 49, the recovered number, and the remainder 0 is less than the divisor 7, so the answer is valid.

Convert to an equation (tool 13): the number N satisfies N = 6 times 8 + 1 = 49, then solve N = 7q + r with 0 less than or equal to r less than 7, giving q = 7, r = 0.

Standards · min grade 3

3.OA.B.6 Understand division as an unknown-factor problem — Rebuilding the number 49 from quotient 8 and remainder 1.
3.OA.C.7 Fluently multiply and divide within 100 — Dividing 49 by 7 to get quotient 7 and remainder 0.

💡 This only needs Grade 3 division: multiply back to find the number, then divide it again!

Variant 6 answer: Quotient 6, remainder 8

When a certain number is divided by $8$ , the quotient is $7$ and the remainder is $6$ . Find the quotient and the remainder when this number is divided by $9$ .

Show solution

Understand

A number divided by 8 gives quotient 7 and remainder 6. We first recover that number, then divide it by 9 to find the new quotient and remainder.

Givens

The number divided by 8 has quotient 7 and remainder 6.
We then divide the same number by 9.

Unknowns

The quotient and remainder when the number is divided by 9.

Constraints

Remainders must be less than the divisor (less than 8 here, less than 9 in the second division).
The number is a whole number.

Plan

#11 Work Backwards · also uses: #6 Guess and Check

Rebuild the original number from quotient times divisor plus remainder, then carry out the new division by 9.

Execute

#11 Work Backwards 3.OA.B.6

The number equals 8 times the quotient 7, plus the remainder 6.

8 \times 7 + 6 = 56 + 6 = 62

Multiplying the quotient by the divisor and adding the remainder undoes the division.

#6 Guess and Check 3.OA.C.7

Now divide 62 by 9. 9 goes into 62 6 times, using 54, and 8 are left over.

62 \div 9 = 6 \cdots 8

Since 9 times 6 is 54 and 62 minus 54 is 8, the quotient is 6 and the remainder 8 (less than 9).

Answer: Quotient 6, remainder 8

Review

Check: 9 times 6 plus 8 equals 62, the recovered number, and the remainder 8 is less than the divisor 9, so the answer is valid.

Convert to an equation (tool 13): the number N satisfies N = 8 times 7 + 6 = 62, then solve N = 9q + r with 0 less than or equal to r less than 9, giving q = 6, r = 8.

Standards · min grade 3

3.OA.B.6 Understand division as an unknown-factor problem — Rebuilding the number 62 from quotient 7 and remainder 6.
3.OA.C.7 Fluently multiply and divide within 100 — Dividing 62 by 9 to get quotient 6 and remainder 8.

💡 This only needs Grade 3 division: multiply back to find the number, then divide it again!

Variant 7 answer: Quotient 8, remainder 1

When a certain number is divided by $7$ , the quotient is $9$ and the remainder is $2$ . Find the quotient and the remainder when this number is divided by $8$ .

Show solution

Understand

A number divided by 7 gives quotient 9 and remainder 2. We first recover that number, then divide it by 8 to find the new quotient and remainder.

Givens

The number divided by 7 has quotient 9 and remainder 2.
We then divide the same number by 8.

Unknowns

The quotient and remainder when the number is divided by 8.

Constraints

Remainders must be less than the divisor (less than 7 here, less than 8 in the second division).
The number is a whole number.

Plan

#11 Work Backwards · also uses: #6 Guess and Check

Rebuild the original number from quotient times divisor plus remainder, then carry out the new division by 8.

Execute

#11 Work Backwards 3.OA.B.6

The number equals 7 times the quotient 9, plus the remainder 2.

7 \times 9 + 2 = 63 + 2 = 65

Multiplying the quotient by the divisor and adding the remainder undoes the division.

#6 Guess and Check 3.OA.C.7

Now divide 65 by 8. 8 goes into 65 8 times, using 64, and 1 are left over.

65 \div 8 = 8 \cdots 1

Since 8 times 8 is 64 and 65 minus 64 is 1, the quotient is 8 and the remainder 1 (less than 8).

Answer: Quotient 8, remainder 1

Review

Check: 8 times 8 plus 1 equals 65, the recovered number, and the remainder 1 is less than the divisor 8, so the answer is valid.

Convert to an equation (tool 13): the number N satisfies N = 7 times 9 + 2 = 65, then solve N = 8q + r with 0 less than or equal to r less than 8, giving q = 8, r = 1.

Standards · min grade 3

3.OA.B.6 Understand division as an unknown-factor problem — Rebuilding the number 65 from quotient 9 and remainder 2.
3.OA.C.7 Fluently multiply and divide within 100 — Dividing 65 by 8 to get quotient 8 and remainder 1.

💡 This only needs Grade 3 division: multiply back to find the number, then divide it again!

Variant 8 answer: Quotient 7, remainder 4

When a certain number is divided by $4$ , the quotient is $9$ and the remainder is $3$ . Find the quotient and the remainder when this number is divided by $5$ .

Show solution

Understand

A number divided by 4 gives quotient 9 and remainder 3. We first recover that number, then divide it by 5 to find the new quotient and remainder.

Givens

The number divided by 4 has quotient 9 and remainder 3.
We then divide the same number by 5.

Unknowns

The quotient and remainder when the number is divided by 5.

Constraints

Remainders must be less than the divisor (less than 4 here, less than 5 in the second division).
The number is a whole number.

Plan

#11 Work Backwards · also uses: #6 Guess and Check

Rebuild the original number from quotient times divisor plus remainder, then carry out the new division by 5.

Execute

#11 Work Backwards 3.OA.B.6

The number equals 4 times the quotient 9, plus the remainder 3.

4 \times 9 + 3 = 36 + 3 = 39

Multiplying the quotient by the divisor and adding the remainder undoes the division.

#6 Guess and Check 3.OA.C.7

Now divide 39 by 5. 5 goes into 39 7 times, using 35, and 4 are left over.

39 \div 5 = 7 \cdots 4

Since 5 times 7 is 35 and 39 minus 35 is 4, the quotient is 7 and the remainder 4 (less than 5).

Answer: Quotient 7, remainder 4

Review

Check: 5 times 7 plus 4 equals 39, the recovered number, and the remainder 4 is less than the divisor 5, so the answer is valid.

Convert to an equation (tool 13): the number N satisfies N = 4 times 9 + 3 = 39, then solve N = 5q + r with 0 less than or equal to r less than 5, giving q = 7, r = 4.

Standards · min grade 3

3.OA.B.6 Understand division as an unknown-factor problem — Rebuilding the number 39 from quotient 9 and remainder 3.
3.OA.C.7 Fluently multiply and divide within 100 — Dividing 39 by 5 to get quotient 7 and remainder 4.

💡 This only needs Grade 3 division: multiply back to find the number, then divide it again!

Variant 9 answer: Quotient 10, remainder 5

When a certain number is divided by $9$ , the quotient is $8$ and the remainder is $3$ . Find the quotient and the remainder when this number is divided by $7$ .

Show solution

Understand

A number divided by 9 gives quotient 8 and remainder 3. We first recover that number, then divide it by 7 to find the new quotient and remainder.

Givens

The number divided by 9 has quotient 8 and remainder 3.
We then divide the same number by 7.

Unknowns

The quotient and remainder when the number is divided by 7.

Constraints

Remainders must be less than the divisor (less than 9 here, less than 7 in the second division).
The number is a whole number.

Plan

#11 Work Backwards · also uses: #6 Guess and Check

Rebuild the original number from quotient times divisor plus remainder, then carry out the new division by 7.

Execute

#11 Work Backwards 3.OA.B.6

The number equals 9 times the quotient 8, plus the remainder 3.

9 \times 8 + 3 = 72 + 3 = 75

Multiplying the quotient by the divisor and adding the remainder undoes the division.

#6 Guess and Check 3.OA.C.7

Now divide 75 by 7. 7 goes into 75 10 times, using 70, and 5 are left over.

75 \div 7 = 10 \cdots 5

Since 7 times 10 is 70 and 75 minus 70 is 5, the quotient is 10 and the remainder 5 (less than 7).

Answer: Quotient 10, remainder 5

Review

Check: 7 times 10 plus 5 equals 75, the recovered number, and the remainder 5 is less than the divisor 7, so the answer is valid.

Convert to an equation (tool 13): the number N satisfies N = 9 times 8 + 3 = 75, then solve N = 7q + r with 0 less than or equal to r less than 7, giving q = 10, r = 5.

Standards · min grade 3

3.OA.B.6 Understand division as an unknown-factor problem — Rebuilding the number 75 from quotient 8 and remainder 3.
3.OA.C.7 Fluently multiply and divide within 100 — Dividing 75 by 7 to get quotient 10 and remainder 5.

💡 This only needs Grade 3 division: multiply back to find the number, then divide it again!

Variant 10 answer: Quotient 14, remainder 3

When a certain number is divided by $8$ , the quotient is $9$ and the remainder is $1$ . Find the quotient and the remainder when this number is divided by $5$ .

Show solution

Understand

A number divided by 8 gives quotient 9 and remainder 1. We first recover that number, then divide it by 5 to find the new quotient and remainder.

Givens

The number divided by 8 has quotient 9 and remainder 1.
We then divide the same number by 5.

Unknowns

The quotient and remainder when the number is divided by 5.

Constraints

Remainders must be less than the divisor (less than 8 here, less than 5 in the second division).
The number is a whole number.

Plan

#11 Work Backwards · also uses: #6 Guess and Check

Rebuild the original number from quotient times divisor plus remainder, then carry out the new division by 5.

Execute

#11 Work Backwards 3.OA.B.6

The number equals 8 times the quotient 9, plus the remainder 1.

8 \times 9 + 1 = 72 + 1 = 73

Multiplying the quotient by the divisor and adding the remainder undoes the division.

#6 Guess and Check 3.OA.C.7

Now divide 73 by 5. 5 goes into 73 14 times, using 70, and 3 are left over.

73 \div 5 = 14 \cdots 3

Since 5 times 14 is 70 and 73 minus 70 is 3, the quotient is 14 and the remainder 3 (less than 5).

Answer: Quotient 14, remainder 3

Review

Check: 5 times 14 plus 3 equals 73, the recovered number, and the remainder 3 is less than the divisor 5, so the answer is valid.

Convert to an equation (tool 13): the number N satisfies N = 8 times 9 + 1 = 73, then solve N = 5q + r with 0 less than or equal to r less than 5, giving q = 14, r = 3.

Standards · min grade 3

3.OA.B.6 Understand division as an unknown-factor problem — Rebuilding the number 73 from quotient 9 and remainder 1.
3.OA.C.7 Fluently multiply and divide within 100 — Dividing 73 by 5 to get quotient 14 and remainder 3.

💡 This only needs Grade 3 division: multiply back to find the number, then divide it again!

Variant 11 answer: Quotient 14, remainder 1

When a certain number is divided by $9$ , the quotient is $7$ and the remainder is $8$ . Find the quotient and the remainder when this number is divided by $5$ .

Show solution

Understand

A number divided by 9 gives quotient 7 and remainder 8. We first recover that number, then divide it by 5 to find the new quotient and remainder.

Givens

The number divided by 9 has quotient 7 and remainder 8.
We then divide the same number by 5.

Unknowns

The quotient and remainder when the number is divided by 5.

Constraints

Remainders must be less than the divisor (less than 9 here, less than 5 in the second division).
The number is a whole number.

Plan

#11 Work Backwards · also uses: #6 Guess and Check

Rebuild the original number from quotient times divisor plus remainder, then carry out the new division by 5.

Execute

#11 Work Backwards 3.OA.B.6

The number equals 9 times the quotient 7, plus the remainder 8.

9 \times 7 + 8 = 63 + 8 = 71

Multiplying the quotient by the divisor and adding the remainder undoes the division.

#6 Guess and Check 3.OA.C.7

Now divide 71 by 5. 5 goes into 71 14 times, using 70, and 1 are left over.

71 \div 5 = 14 \cdots 1

Since 5 times 14 is 70 and 71 minus 70 is 1, the quotient is 14 and the remainder 1 (less than 5).

Answer: Quotient 14, remainder 1

Review

Check: 5 times 14 plus 1 equals 71, the recovered number, and the remainder 1 is less than the divisor 5, so the answer is valid.

Convert to an equation (tool 13): the number N satisfies N = 9 times 7 + 8 = 71, then solve N = 5q + r with 0 less than or equal to r less than 5, giving q = 14, r = 1.

Standards · min grade 3

3.OA.B.6 Understand division as an unknown-factor problem — Rebuilding the number 71 from quotient 7 and remainder 8.
3.OA.C.7 Fluently multiply and divide within 100 — Dividing 71 by 5 to get quotient 14 and remainder 1.

💡 This only needs Grade 3 division: multiply back to find the number, then divide it again!

Variant 12 answer: Quotient 6, remainder 1

When a certain number is divided by $8$ , the quotient is $6$ and the remainder is $7$ . Find the quotient and the remainder when this number is divided by $9$ .

Show solution

Understand

A number divided by 8 gives quotient 6 and remainder 7. We first recover that number, then divide it by 9 to find the new quotient and remainder.

Givens

The number divided by 8 has quotient 6 and remainder 7.
We then divide the same number by 9.

Unknowns

The quotient and remainder when the number is divided by 9.

Constraints

Remainders must be less than the divisor (less than 8 here, less than 9 in the second division).
The number is a whole number.

Plan

#11 Work Backwards · also uses: #6 Guess and Check

Rebuild the original number from quotient times divisor plus remainder, then carry out the new division by 9.

Execute

#11 Work Backwards 3.OA.B.6

The number equals 8 times the quotient 6, plus the remainder 7.

8 \times 6 + 7 = 48 + 7 = 55

Multiplying the quotient by the divisor and adding the remainder undoes the division.

#6 Guess and Check 3.OA.C.7

Now divide 55 by 9. 9 goes into 55 6 times, using 54, and 1 are left over.

55 \div 9 = 6 \cdots 1

Since 9 times 6 is 54 and 55 minus 54 is 1, the quotient is 6 and the remainder 1 (less than 9).

Answer: Quotient 6, remainder 1

Review

Check: 9 times 6 plus 1 equals 55, the recovered number, and the remainder 1 is less than the divisor 9, so the answer is valid.

Convert to an equation (tool 13): the number N satisfies N = 8 times 6 + 7 = 55, then solve N = 9q + r with 0 less than or equal to r less than 9, giving q = 6, r = 1.

Standards · min grade 3

3.OA.B.6 Understand division as an unknown-factor problem — Rebuilding the number 55 from quotient 6 and remainder 7.
3.OA.C.7 Fluently multiply and divide within 100 — Dividing 55 by 9 to get quotient 6 and remainder 1.

💡 This only needs Grade 3 division: multiply back to find the number, then divide it again!