← Remainder must be less than the divisor · Divisibility and Remainder Reasoning

Remainder must be less than the divisor · 12 practice problems

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

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Generated variants — 12

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

Variant 1 easy

Find the greatest number that $\blacksquare$ can be.

$\blacksquare \div 8 = 6 \cdots \blacktriangle$

Your answer

Show solution

1 · Understandwhat's really being asked

A number (the blank) divided by 8 gives quotient 6 and some remainder (the triangle). We want the greatest possible value of the blank, so we make the remainder as large as it can be.

Givens

The blank divided by 8 has quotient 6 and remainder equal to the triangle.
The remainder is whatever makes the blank largest.

Unknowns

The greatest possible value of the blank (the dividend).

Constraints

The remainder must be less than the divisor 8.
The quotient stays exactly 6.

2 · Planchoose the strategy

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

Use dividend = divisor times quotient plus remainder. To make the dividend largest while keeping quotient 6, choose the biggest allowed remainder, which is 7.

3 · Execute2 carry out the plan

1Pick the largest remainder

#6 Guess and Check 3.OA.B.6

A remainder must be smaller than the divisor 8, so the greatest it can be is 7. A remainder of 8 or more would increase the quotient.

\blacktriangle < 8 \Rightarrow \blacktriangle = 7

The remainder is always less than the divisor, so 7 is the biggest value that keeps the quotient at 6.

2Build the greatest dividend

#11 Work Backwards 3.OA.C.7

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

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

Multiplying the divisor by the quotient and adding the largest remainder gives the largest number that still divides to give 6.

Answer: 55

4 · Reviewdoes it hold up?

Check: 55 divided by 8 is 6 remainder 7, since 8 times 6 is 48 and 55 minus 48 is 7, and 7 is less than 8. Trying 56 would give quotient 7, too big, so 55 is the greatest.

Another way: List the candidates (tool 2): dividends giving quotient 6 are 48 (r 0) up to 55 (r 7); the next, 56, bumps the quotient to 7, so 55 is the maximum.

Standardsmin grade 3

3.OA.B.6 Understand division as an unknown-factor problem — Recognizing the remainder must be less than 8, so its largest value is 7.
3.OA.C.7 Fluently multiply and divide within 100 — Computing 8 times 6 plus 7 to get the greatest dividend.

💡Takeaway. This only needs Grade 3 division: the remainder maxes out at one less than the divisor, then multiply back!

Variant 2 easy

Find the greatest number that $\blacksquare$ can be.

$\blacksquare \div 7 = 8 \cdots \blacktriangle$

Your answer

Show solution

1 · Understandwhat's really being asked

A number (the blank) divided by 7 gives quotient 8 and some remainder (the triangle). We want the greatest possible value of the blank, so we make the remainder as large as it can be.

Givens

The blank divided by 7 has quotient 8 and remainder equal to the triangle.
The remainder is whatever makes the blank largest.

Unknowns

The greatest possible value of the blank (the dividend).

Constraints

The remainder must be less than the divisor 7.
The quotient stays exactly 8.

2 · Planchoose the strategy

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

Use dividend = divisor times quotient plus remainder. To make the dividend largest while keeping quotient 8, choose the biggest allowed remainder, which is 6.

3 · Execute2 carry out the plan

1Pick the largest remainder

#6 Guess and Check 3.OA.B.6

A remainder must be smaller than the divisor 7, so the greatest it can be is 6. A remainder of 7 or more would increase the quotient.

\blacktriangle < 7 \Rightarrow \blacktriangle = 6

The remainder is always less than the divisor, so 6 is the biggest value that keeps the quotient at 8.

2Build the greatest dividend

#11 Work Backwards 3.OA.C.7

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

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

Multiplying the divisor by the quotient and adding the largest remainder gives the largest number that still divides to give 8.

Answer: 62

4 · Reviewdoes it hold up?

Check: 62 divided by 7 is 8 remainder 6, since 7 times 8 is 56 and 62 minus 56 is 6, and 6 is less than 7. Trying 63 would give quotient 9, too big, so 62 is the greatest.

Another way: List the candidates (tool 2): dividends giving quotient 8 are 56 (r 0) up to 62 (r 6); the next, 63, bumps the quotient to 9, so 62 is the maximum.

Standardsmin grade 3

3.OA.B.6 Understand division as an unknown-factor problem — Recognizing the remainder must be less than 7, so its largest value is 6.
3.OA.C.7 Fluently multiply and divide within 100 — Computing 7 times 8 plus 6 to get the greatest dividend.

💡Takeaway. This only needs Grade 3 division: the remainder maxes out at one less than the divisor, then multiply back!

Variant 3 medium

Find the greatest number that $\blacksquare$ can be.

$\blacksquare \div 8 = 9 \cdots \blacktriangle$

Your answer

Show solution

1 · Understandwhat's really being asked

A number (the blank) divided by 8 gives quotient 9 and some remainder (the triangle). We want the greatest possible value of the blank, so we make the remainder as large as it can be.

Givens

The blank divided by 8 has quotient 9 and remainder equal to the triangle.
The remainder is whatever makes the blank largest.

Unknowns

The greatest possible value of the blank (the dividend).

Constraints

The remainder must be less than the divisor 8.
The quotient stays exactly 9.

2 · Planchoose the strategy

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

Use dividend = divisor times quotient plus remainder. To make the dividend largest while keeping quotient 9, choose the biggest allowed remainder, which is 7.

3 · Execute2 carry out the plan

1Pick the largest remainder

#6 Guess and Check 3.OA.B.6

A remainder must be smaller than the divisor 8, so the greatest it can be is 7. A remainder of 8 or more would increase the quotient.

\blacktriangle < 8 \Rightarrow \blacktriangle = 7

The remainder is always less than the divisor, so 7 is the biggest value that keeps the quotient at 9.

2Build the greatest dividend

#11 Work Backwards 3.OA.C.7

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

8 \times 9 + 7 = 72 + 7 = 79

Multiplying the divisor by the quotient and adding the largest remainder gives the largest number that still divides to give 9.

Answer: 79

4 · Reviewdoes it hold up?

Check: 79 divided by 8 is 9 remainder 7, since 8 times 9 is 72 and 79 minus 72 is 7, and 7 is less than 8. Trying 80 would give quotient 10, too big, so 79 is the greatest.

Another way: List the candidates (tool 2): dividends giving quotient 9 are 72 (r 0) up to 79 (r 7); the next, 80, bumps the quotient to 10, so 79 is the maximum.

Standardsmin grade 3

3.OA.B.6 Understand division as an unknown-factor problem — Recognizing the remainder must be less than 8, so its largest value is 7.
3.OA.C.7 Fluently multiply and divide within 100 — Computing 8 times 9 plus 7 to get the greatest dividend.

💡Takeaway. This only needs Grade 3 division: the remainder maxes out at one less than the divisor, then multiply back!

Variant 4 easy

Find the greatest number that $\blacksquare$ can be.

$\blacksquare \div 9 = 7 \cdots \blacktriangle$

Your answer

Show solution

1 · Understandwhat's really being asked

A number (the blank) divided by 9 gives quotient 7 and some remainder (the triangle). We want the greatest possible value of the blank, so we make the remainder as large as it can be.

Givens

The blank divided by 9 has quotient 7 and remainder equal to the triangle.
The remainder is whatever makes the blank largest.

Unknowns

The greatest possible value of the blank (the dividend).

Constraints

The remainder must be less than the divisor 9.
The quotient stays exactly 7.

2 · Planchoose the strategy

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

Use dividend = divisor times quotient plus remainder. To make the dividend largest while keeping quotient 7, choose the biggest allowed remainder, which is 8.

3 · Execute2 carry out the plan

1Pick the largest remainder

#6 Guess and Check 3.OA.B.6

A remainder must be smaller than the divisor 9, so the greatest it can be is 8. A remainder of 9 or more would increase the quotient.

\blacktriangle < 9 \Rightarrow \blacktriangle = 8

The remainder is always less than the divisor, so 8 is the biggest value that keeps the quotient at 7.

2Build the greatest dividend

#11 Work Backwards 3.OA.C.7

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

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

Multiplying the divisor by the quotient and adding the largest remainder gives the largest number that still divides to give 7.

Answer: 71

4 · Reviewdoes it hold up?

Check: 71 divided by 9 is 7 remainder 8, since 9 times 7 is 63 and 71 minus 63 is 8, and 8 is less than 9. Trying 72 would give quotient 8, too big, so 71 is the greatest.

Another way: List the candidates (tool 2): dividends giving quotient 7 are 63 (r 0) up to 71 (r 8); the next, 72, bumps the quotient to 8, so 71 is the maximum.

Standardsmin grade 3

3.OA.B.6 Understand division as an unknown-factor problem — Recognizing the remainder must be less than 9, so its largest value is 8.
3.OA.C.7 Fluently multiply and divide within 100 — Computing 9 times 7 plus 8 to get the greatest dividend.

💡Takeaway. This only needs Grade 3 division: the remainder maxes out at one less than the divisor, then multiply back!

Variant 5 medium

Find the greatest number that $\blacksquare$ can be.

$\blacksquare \div 9 = 9 \cdots \blacktriangle$

Your answer

Show solution

1 · Understandwhat's really being asked

A number (the blank) divided by 9 gives quotient 9 and some remainder (the triangle). We want the greatest possible value of the blank, so we make the remainder as large as it can be.

Givens

The blank divided by 9 has quotient 9 and remainder equal to the triangle.
The remainder is whatever makes the blank largest.

Unknowns

The greatest possible value of the blank (the dividend).

Constraints

The remainder must be less than the divisor 9.
The quotient stays exactly 9.

2 · Planchoose the strategy

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

Use dividend = divisor times quotient plus remainder. To make the dividend largest while keeping quotient 9, choose the biggest allowed remainder, which is 8.

3 · Execute2 carry out the plan

1Pick the largest remainder

#6 Guess and Check 3.OA.B.6

A remainder must be smaller than the divisor 9, so the greatest it can be is 8. A remainder of 9 or more would increase the quotient.

\blacktriangle < 9 \Rightarrow \blacktriangle = 8

The remainder is always less than the divisor, so 8 is the biggest value that keeps the quotient at 9.

2Build the greatest dividend

#11 Work Backwards 3.OA.C.7

The dividend equals 9 times the quotient 9, plus the remainder 8.

9 \times 9 + 8 = 81 + 8 = 89

Multiplying the divisor by the quotient and adding the largest remainder gives the largest number that still divides to give 9.

Answer: 89

4 · Reviewdoes it hold up?

Check: 89 divided by 9 is 9 remainder 8, since 9 times 9 is 81 and 89 minus 81 is 8, and 8 is less than 9. Trying 90 would give quotient 10, too big, so 89 is the greatest.

Another way: List the candidates (tool 2): dividends giving quotient 9 are 81 (r 0) up to 89 (r 8); the next, 90, bumps the quotient to 10, so 89 is the maximum.

Standardsmin grade 3

3.OA.B.6 Understand division as an unknown-factor problem — Recognizing the remainder must be less than 9, so its largest value is 8.
3.OA.C.7 Fluently multiply and divide within 100 — Computing 9 times 9 plus 8 to get the greatest dividend.

💡Takeaway. This only needs Grade 3 division: the remainder maxes out at one less than the divisor, then multiply back!

Variant 6 easy

Find the greatest number that $\blacksquare$ can be.

$\blacksquare \div 6 = 9 \cdots \blacktriangle$

Your answer

Show solution

1 · Understandwhat's really being asked

A number (the blank) divided by 6 gives quotient 9 and some remainder (the triangle). We want the greatest possible value of the blank, so we make the remainder as large as it can be.

Givens

The blank divided by 6 has quotient 9 and remainder equal to the triangle.
The remainder is whatever makes the blank largest.

Unknowns

The greatest possible value of the blank (the dividend).

Constraints

The remainder must be less than the divisor 6.
The quotient stays exactly 9.

2 · Planchoose the strategy

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

Use dividend = divisor times quotient plus remainder. To make the dividend largest while keeping quotient 9, choose the biggest allowed remainder, which is 5.

3 · Execute2 carry out the plan

1Pick the largest remainder

#6 Guess and Check 3.OA.B.6

A remainder must be smaller than the divisor 6, so the greatest it can be is 5. A remainder of 6 or more would increase the quotient.

\blacktriangle < 6 \Rightarrow \blacktriangle = 5

The remainder is always less than the divisor, so 5 is the biggest value that keeps the quotient at 9.

2Build the greatest dividend

#11 Work Backwards 3.OA.C.7

The dividend equals 6 times the quotient 9, plus the remainder 5.

6 \times 9 + 5 = 54 + 5 = 59

Multiplying the divisor by the quotient and adding the largest remainder gives the largest number that still divides to give 9.

Answer: 59

4 · Reviewdoes it hold up?

Check: 59 divided by 6 is 9 remainder 5, since 6 times 9 is 54 and 59 minus 54 is 5, and 5 is less than 6. Trying 60 would give quotient 10, too big, so 59 is the greatest.

Another way: List the candidates (tool 2): dividends giving quotient 9 are 54 (r 0) up to 59 (r 5); the next, 60, bumps the quotient to 10, so 59 is the maximum.

Standardsmin grade 3

3.OA.B.6 Understand division as an unknown-factor problem — Recognizing the remainder must be less than 6, so its largest value is 5.
3.OA.C.7 Fluently multiply and divide within 100 — Computing 6 times 9 plus 5 to get the greatest dividend.

💡Takeaway. This only needs Grade 3 division: the remainder maxes out at one less than the divisor, then multiply back!

Variant 7 medium

Find the greatest number that $\blacksquare$ can be.

$\blacksquare \div 6 = 10 \cdots \blacktriangle$

Your answer

Show solution

1 · Understandwhat's really being asked

A number (the blank) divided by 6 gives quotient 10 and some remainder (the triangle). We want the greatest possible value of the blank, so we make the remainder as large as it can be.

Givens

The blank divided by 6 has quotient 10 and remainder equal to the triangle.
The remainder is whatever makes the blank largest.

Unknowns

The greatest possible value of the blank (the dividend).

Constraints

The remainder must be less than the divisor 6.
The quotient stays exactly 10.

2 · Planchoose the strategy

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

Use dividend = divisor times quotient plus remainder. To make the dividend largest while keeping quotient 10, choose the biggest allowed remainder, which is 5.

3 · Execute2 carry out the plan

1Pick the largest remainder

#6 Guess and Check 3.OA.B.6

A remainder must be smaller than the divisor 6, so the greatest it can be is 5. A remainder of 6 or more would increase the quotient.

\blacktriangle < 6 \Rightarrow \blacktriangle = 5

The remainder is always less than the divisor, so 5 is the biggest value that keeps the quotient at 10.

2Build the greatest dividend

#11 Work Backwards 3.OA.C.7

The dividend equals 6 times the quotient 10, plus the remainder 5.

6 \times 10 + 5 = 60 + 5 = 65

Multiplying the divisor by the quotient and adding the largest remainder gives the largest number that still divides to give 10.

Answer: 65

4 · Reviewdoes it hold up?

Check: 65 divided by 6 is 10 remainder 5, since 6 times 10 is 60 and 65 minus 60 is 5, and 5 is less than 6. Trying 66 would give quotient 11, too big, so 65 is the greatest.

Another way: List the candidates (tool 2): dividends giving quotient 10 are 60 (r 0) up to 65 (r 5); the next, 66, bumps the quotient to 11, so 65 is the maximum.

Standardsmin grade 3

3.OA.B.6 Understand division as an unknown-factor problem — Recognizing the remainder must be less than 6, so its largest value is 5.
3.OA.C.7 Fluently multiply and divide within 100 — Computing 6 times 10 plus 5 to get the greatest dividend.

💡Takeaway. This only needs Grade 3 division: the remainder maxes out at one less than the divisor, then multiply back!

Variant 8 hard

Find the greatest number that $\blacksquare$ can be.

$\blacksquare \div 7 = 11 \cdots \blacktriangle$

Your answer

Show solution

1 · Understandwhat's really being asked

A number (the blank) divided by 7 gives quotient 11 and some remainder (the triangle). We want the greatest possible value of the blank, so we make the remainder as large as it can be.

Givens

The blank divided by 7 has quotient 11 and remainder equal to the triangle.
The remainder is whatever makes the blank largest.

Unknowns

The greatest possible value of the blank (the dividend).

Constraints

The remainder must be less than the divisor 7.
The quotient stays exactly 11.

2 · Planchoose the strategy

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

Use dividend = divisor times quotient plus remainder. To make the dividend largest while keeping quotient 11, choose the biggest allowed remainder, which is 6.

3 · Execute2 carry out the plan

1Pick the largest remainder

#6 Guess and Check 3.OA.B.6

A remainder must be smaller than the divisor 7, so the greatest it can be is 6. A remainder of 7 or more would increase the quotient.

\blacktriangle < 7 \Rightarrow \blacktriangle = 6

The remainder is always less than the divisor, so 6 is the biggest value that keeps the quotient at 11.

2Build the greatest dividend

#11 Work Backwards 3.OA.C.7

The dividend equals 7 times the quotient 11, plus the remainder 6.

7 \times 11 + 6 = 77 + 6 = 83

Multiplying the divisor by the quotient and adding the largest remainder gives the largest number that still divides to give 11.

Answer: 83

4 · Reviewdoes it hold up?

Check: 83 divided by 7 is 11 remainder 6, since 7 times 11 is 77 and 83 minus 77 is 6, and 6 is less than 7. Trying 84 would give quotient 12, too big, so 83 is the greatest.

Another way: List the candidates (tool 2): dividends giving quotient 11 are 77 (r 0) up to 83 (r 6); the next, 84, bumps the quotient to 12, so 83 is the maximum.

Standardsmin grade 3

3.OA.B.6 Understand division as an unknown-factor problem — Recognizing the remainder must be less than 7, so its largest value is 6.
3.OA.C.7 Fluently multiply and divide within 100 — Computing 7 times 11 plus 6 to get the greatest dividend.

💡Takeaway. This only needs Grade 3 division: the remainder maxes out at one less than the divisor, then multiply back!

Variant 9 medium

Find the greatest number that $\blacksquare$ can be.

$\blacksquare \div 8 = 11 \cdots \blacktriangle$

Your answer

Show solution

1 · Understandwhat's really being asked

A number (the blank) divided by 8 gives quotient 11 and some remainder (the triangle). We want the greatest possible value of the blank, so we make the remainder as large as it can be.

Givens

The blank divided by 8 has quotient 11 and remainder equal to the triangle.
The remainder is whatever makes the blank largest.

Unknowns

The greatest possible value of the blank (the dividend).

Constraints

The remainder must be less than the divisor 8.
The quotient stays exactly 11.

2 · Planchoose the strategy

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

Use dividend = divisor times quotient plus remainder. To make the dividend largest while keeping quotient 11, choose the biggest allowed remainder, which is 7.

3 · Execute2 carry out the plan

1Pick the largest remainder

#6 Guess and Check 3.OA.B.6

A remainder must be smaller than the divisor 8, so the greatest it can be is 7. A remainder of 8 or more would increase the quotient.

\blacktriangle < 8 \Rightarrow \blacktriangle = 7

The remainder is always less than the divisor, so 7 is the biggest value that keeps the quotient at 11.

2Build the greatest dividend

#11 Work Backwards 3.OA.C.7

The dividend equals 8 times the quotient 11, plus the remainder 7.

8 \times 11 + 7 = 88 + 7 = 95

Multiplying the divisor by the quotient and adding the largest remainder gives the largest number that still divides to give 11.

Answer: 95

4 · Reviewdoes it hold up?

Check: 95 divided by 8 is 11 remainder 7, since 8 times 11 is 88 and 95 minus 88 is 7, and 7 is less than 8. Trying 96 would give quotient 12, too big, so 95 is the greatest.

Another way: List the candidates (tool 2): dividends giving quotient 11 are 88 (r 0) up to 95 (r 7); the next, 96, bumps the quotient to 12, so 95 is the maximum.

Standardsmin grade 3

3.OA.B.6 Understand division as an unknown-factor problem — Recognizing the remainder must be less than 8, so its largest value is 7.
3.OA.C.7 Fluently multiply and divide within 100 — Computing 8 times 11 plus 7 to get the greatest dividend.

💡Takeaway. This only needs Grade 3 division: the remainder maxes out at one less than the divisor, then multiply back!

Variant 10 hard

Find the greatest number that $\blacksquare$ can be.

$\blacksquare \div 5 = 12 \cdots \blacktriangle$

Your answer

Show solution

1 · Understandwhat's really being asked

A number (the blank) divided by 5 gives quotient 12 and some remainder (the triangle). We want the greatest possible value of the blank, so we make the remainder as large as it can be.

Givens

The blank divided by 5 has quotient 12 and remainder equal to the triangle.
The remainder is whatever makes the blank largest.

Unknowns

The greatest possible value of the blank (the dividend).

Constraints

The remainder must be less than the divisor 5.
The quotient stays exactly 12.

2 · Planchoose the strategy

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

Use dividend = divisor times quotient plus remainder. To make the dividend largest while keeping quotient 12, choose the biggest allowed remainder, which is 4.

3 · Execute2 carry out the plan

1Pick the largest remainder

#6 Guess and Check 3.OA.B.6

A remainder must be smaller than the divisor 5, so the greatest it can be is 4. A remainder of 5 or more would increase the quotient.

\blacktriangle < 5 \Rightarrow \blacktriangle = 4

The remainder is always less than the divisor, so 4 is the biggest value that keeps the quotient at 12.

2Build the greatest dividend

#11 Work Backwards 3.OA.C.7

The dividend equals 5 times the quotient 12, plus the remainder 4.

5 \times 12 + 4 = 60 + 4 = 64

Multiplying the divisor by the quotient and adding the largest remainder gives the largest number that still divides to give 12.

Answer: 64

4 · Reviewdoes it hold up?

Check: 64 divided by 5 is 12 remainder 4, since 5 times 12 is 60 and 64 minus 60 is 4, and 4 is less than 5. Trying 65 would give quotient 13, too big, so 64 is the greatest.

Another way: List the candidates (tool 2): dividends giving quotient 12 are 60 (r 0) up to 64 (r 4); the next, 65, bumps the quotient to 13, so 64 is the maximum.

Standardsmin grade 3

3.OA.B.6 Understand division as an unknown-factor problem — Recognizing the remainder must be less than 5, so its largest value is 4.
3.OA.C.7 Fluently multiply and divide within 100 — Computing 5 times 12 plus 4 to get the greatest dividend.

💡Takeaway. This only needs Grade 3 division: the remainder maxes out at one less than the divisor, then multiply back!

Variant 11 hard

Find the greatest number that $\blacksquare$ can be.

$\blacksquare \div 5 = 14 \cdots \blacktriangle$

Your answer

Show solution

1 · Understandwhat's really being asked

A number (the blank) divided by 5 gives quotient 14 and some remainder (the triangle). We want the greatest possible value of the blank, so we make the remainder as large as it can be.

Givens

The blank divided by 5 has quotient 14 and remainder equal to the triangle.
The remainder is whatever makes the blank largest.

Unknowns

The greatest possible value of the blank (the dividend).

Constraints

The remainder must be less than the divisor 5.
The quotient stays exactly 14.

2 · Planchoose the strategy

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

Use dividend = divisor times quotient plus remainder. To make the dividend largest while keeping quotient 14, choose the biggest allowed remainder, which is 4.

3 · Execute2 carry out the plan

1Pick the largest remainder

#6 Guess and Check 3.OA.B.6

A remainder must be smaller than the divisor 5, so the greatest it can be is 4. A remainder of 5 or more would increase the quotient.

\blacktriangle < 5 \Rightarrow \blacktriangle = 4

The remainder is always less than the divisor, so 4 is the biggest value that keeps the quotient at 14.

2Build the greatest dividend

#11 Work Backwards 3.OA.C.7

The dividend equals 5 times the quotient 14, plus the remainder 4.

5 \times 14 + 4 = 70 + 4 = 74

Multiplying the divisor by the quotient and adding the largest remainder gives the largest number that still divides to give 14.

Answer: 74

4 · Reviewdoes it hold up?

Check: 74 divided by 5 is 14 remainder 4, since 5 times 14 is 70 and 74 minus 70 is 4, and 4 is less than 5. Trying 75 would give quotient 15, too big, so 74 is the greatest.

Another way: List the candidates (tool 2): dividends giving quotient 14 are 70 (r 0) up to 74 (r 4); the next, 75, bumps the quotient to 15, so 74 is the maximum.

Standardsmin grade 3

3.OA.B.6 Understand division as an unknown-factor problem — Recognizing the remainder must be less than 5, so its largest value is 4.
3.OA.C.7 Fluently multiply and divide within 100 — Computing 5 times 14 plus 4 to get the greatest dividend.

💡Takeaway. This only needs Grade 3 division: the remainder maxes out at one less than the divisor, then multiply back!

Variant 12 hard

Find the greatest number that $\blacksquare$ can be.

$\blacksquare \div 4 = 15 \cdots \blacktriangle$

Your answer

Show solution

1 · Understandwhat's really being asked

A number (the blank) divided by 4 gives quotient 15 and some remainder (the triangle). We want the greatest possible value of the blank, so we make the remainder as large as it can be.

Givens

The blank divided by 4 has quotient 15 and remainder equal to the triangle.
The remainder is whatever makes the blank largest.

Unknowns

The greatest possible value of the blank (the dividend).

Constraints

The remainder must be less than the divisor 4.
The quotient stays exactly 15.

2 · Planchoose the strategy

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

Use dividend = divisor times quotient plus remainder. To make the dividend largest while keeping quotient 15, choose the biggest allowed remainder, which is 3.

3 · Execute2 carry out the plan

1Pick the largest remainder

#6 Guess and Check 3.OA.B.6

A remainder must be smaller than the divisor 4, so the greatest it can be is 3. A remainder of 4 or more would increase the quotient.

\blacktriangle < 4 \Rightarrow \blacktriangle = 3

The remainder is always less than the divisor, so 3 is the biggest value that keeps the quotient at 15.

2Build the greatest dividend

#11 Work Backwards 3.OA.C.7

The dividend equals 4 times the quotient 15, plus the remainder 3.

4 \times 15 + 3 = 60 + 3 = 63

Multiplying the divisor by the quotient and adding the largest remainder gives the largest number that still divides to give 15.

Answer: 63

4 · Reviewdoes it hold up?

Check: 63 divided by 4 is 15 remainder 3, since 4 times 15 is 60 and 63 minus 60 is 3, and 3 is less than 4. Trying 64 would give quotient 16, too big, so 63 is the greatest.

Another way: List the candidates (tool 2): dividends giving quotient 15 are 60 (r 0) up to 63 (r 3); the next, 64, bumps the quotient to 16, so 63 is the maximum.

Standardsmin grade 3

3.OA.B.6 Understand division as an unknown-factor problem — Recognizing the remainder must be less than 4, so its largest value is 3.
3.OA.C.7 Fluently multiply and divide within 100 — Computing 4 times 15 plus 3 to get the greatest dividend.

💡Takeaway. This only needs Grade 3 division: the remainder maxes out at one less than the divisor, then multiply back!

Where next?

← Back to the problem: Remainder must be less than the divisor