← 3-2 · Exact division means remainder zero · Divisibility and Remainder Reasoning

Exact division means remainder zero · 11 practice problems

3.OA.C.73.OA.B.6

Generated variants — 11

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

Variant 1 answer: 4

The division below comes out exactly (no remainder). Find every digit that $\blacksquare$ could be. (Here $8\blacksquare$ is a two-digit number.)

$8\blacksquare \div 6$

Show solution

Understand

The two-digit number 8-followed-by-a-blank must divide exactly by 6. We need every digit the blank could be so that the number is a multiple of 6.

Givens

The number is two digits and starts with 8 (it is 80 through 89).
When divided by 6 it must leave no remainder.

Unknowns

All possible digits for the blank (the ones digit).

Constraints

The blank is a single digit 0 through 9.
The whole number must be divisible by 6.

Plan

#2 Make a Systematic List · also uses: #6 Guess and Check

There are only ten numbers from 80 to 89, so we can list the multiples of 6 in that range and read off which ones-digits work.

Execute

#2 Make a Systematic List 3.OA.C.7

The multiples of 6 that land in the 80s are 84. Any multiple outside 80 to 89 does not have 8 as its tens digit.

6 \times 14 = 84

Listing the multiples of 6 inside 80 to 89 catches every number that divides evenly.

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

Each of 84 starts with 8, so the blank can be 4. The other ones digits leave remainders, so they do not work.

84 \div 6 = 14

Each valid number gives a whole quotient, confirming the digit makes a clean division.

Answer: 4

Review

84 are the only multiples of 6 between 80 and 89 (multiples of 6 are spaced 6 apart, so a ten-wide window holds just a few). Each gives a whole quotient, so the answers are correct.

Test the ten numbers 80 through 89 one at a time, dividing each by 6, and keep the ones with no remainder.

Standards · min grade 3

3.OA.C.7 Fluently multiply and divide within 100 — Listing the multiples of 6 within the 80s.
3.OA.B.6 Understand division as an unknown-factor problem — Checking that each candidate divides by 6 with a whole-number quotient.

💡 This only needs Grade 3 multiplication facts: list the multiples of 6 and see which land in the 80s!

Variant 2 answer: 4

The division below comes out exactly (no remainder). Find every digit that $\blacksquare$ could be. (Here $5\blacksquare$ is a two-digit number.)

$5\blacksquare \div 9$

Show solution

Understand

The two-digit number 5-followed-by-a-blank must divide exactly by 9. We need every digit the blank could be so that the number is a multiple of 9.

Givens

The number is two digits and starts with 5 (it is 50 through 59).
When divided by 9 it must leave no remainder.

Unknowns

All possible digits for the blank (the ones digit).

Constraints

The blank is a single digit 0 through 9.
The whole number must be divisible by 9.

Plan

#2 Make a Systematic List · also uses: #6 Guess and Check

There are only ten numbers from 50 to 59, so we can list the multiples of 9 in that range and read off which ones-digits work.

Execute

#2 Make a Systematic List 3.OA.C.7

The multiples of 9 that land in the 50s are 54. Any multiple outside 50 to 59 does not have 5 as its tens digit.

9 \times 6 = 54

Listing the multiples of 9 inside 50 to 59 catches every number that divides evenly.

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

Each of 54 starts with 5, so the blank can be 4. The other ones digits leave remainders, so they do not work.

54 \div 9 = 6

Each valid number gives a whole quotient, confirming the digit makes a clean division.

Answer: 4

Review

54 are the only multiples of 9 between 50 and 59 (multiples of 9 are spaced 9 apart, so a ten-wide window holds just a few). Each gives a whole quotient, so the answers are correct.

Test the ten numbers 50 through 59 one at a time, dividing each by 9, and keep the ones with no remainder.

Standards · min grade 3

3.OA.C.7 Fluently multiply and divide within 100 — Listing the multiples of 9 within the 50s.
3.OA.B.6 Understand division as an unknown-factor problem — Checking that each candidate divides by 9 with a whole-number quotient.

💡 This only needs Grade 3 multiplication facts: list the multiples of 9 and see which land in the 50s!

Variant 3 answer: 4

The division below comes out exactly (no remainder). Find every digit that $\blacksquare$ could be. (Here $6\blacksquare$ is a two-digit number.)

$6\blacksquare \div 8$

Show solution

Understand

The two-digit number 6-followed-by-a-blank must divide exactly by 8. We need every digit the blank could be so that the number is a multiple of 8.

Givens

The number is two digits and starts with 6 (it is 60 through 69).
When divided by 8 it must leave no remainder.

Unknowns

All possible digits for the blank (the ones digit).

Constraints

The blank is a single digit 0 through 9.
The whole number must be divisible by 8.

Plan

#2 Make a Systematic List · also uses: #6 Guess and Check

There are only ten numbers from 60 to 69, so we can list the multiples of 8 in that range and read off which ones-digits work.

Execute

#2 Make a Systematic List 3.OA.C.7

The multiples of 8 that land in the 60s are 64. Any multiple outside 60 to 69 does not have 6 as its tens digit.

8 \times 8 = 64

Listing the multiples of 8 inside 60 to 69 catches every number that divides evenly.

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

Each of 64 starts with 6, so the blank can be 4. The other ones digits leave remainders, so they do not work.

64 \div 8 = 8

Each valid number gives a whole quotient, confirming the digit makes a clean division.

Answer: 4

Review

64 are the only multiples of 8 between 60 and 69 (multiples of 8 are spaced 8 apart, so a ten-wide window holds just a few). Each gives a whole quotient, so the answers are correct.

Test the ten numbers 60 through 69 one at a time, dividing each by 8, and keep the ones with no remainder.

Standards · min grade 3

3.OA.C.7 Fluently multiply and divide within 100 — Listing the multiples of 8 within the 60s.
3.OA.B.6 Understand division as an unknown-factor problem — Checking that each candidate divides by 8 with a whole-number quotient.

💡 This only needs Grade 3 multiplication facts: list the multiples of 8 and see which land in the 60s!

Variant 4 answer: 1, 4, or 7

The division below comes out exactly (no remainder). Find every digit that $\blacksquare$ could be. (Here $5\blacksquare$ is a two-digit number.)

$5\blacksquare \div 3$

Show solution

Understand

The two-digit number 5-followed-by-a-blank must divide exactly by 3. We need every digit the blank could be so that the number is a multiple of 3.

Givens

The number is two digits and starts with 5 (it is 50 through 59).
When divided by 3 it must leave no remainder.

Unknowns

All possible digits for the blank (the ones digit).

Constraints

The blank is a single digit 0 through 9.
The whole number must be divisible by 3.

Plan

#2 Make a Systematic List · also uses: #6 Guess and Check

There are only ten numbers from 50 to 59, so we can list the multiples of 3 in that range and read off which ones-digits work.

Execute

#2 Make a Systematic List 3.OA.C.7

The multiples of 3 that land in the 50s are 51, 54, and 57. Any multiple outside 50 to 59 does not have 5 as its tens digit.

3 \times 17 = 51,\quad 3 \times 18 = 54,\quad 3 \times 19 = 57

Listing the multiples of 3 inside 50 to 59 catches every number that divides evenly.

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

Each of 51, 54, and 57 starts with 5, so the blank can be 1, 4, or 7. The other ones digits leave remainders, so they do not work.

51 \div 3 = 17,\quad 54 \div 3 = 18,\quad 57 \div 3 = 19

Each valid number gives a whole quotient, confirming the digit makes a clean division.

Answer: 1, 4, or 7

Review

51, 54, and 57 are the only multiples of 3 between 50 and 59 (multiples of 3 are spaced 3 apart, so a ten-wide window holds just a few). Each gives a whole quotient, so the answers are correct.

Test the ten numbers 50 through 59 one at a time, dividing each by 3, and keep the ones with no remainder.

Standards · min grade 3

3.OA.C.7 Fluently multiply and divide within 100 — Listing the multiples of 3 within the 50s.
3.OA.B.6 Understand division as an unknown-factor problem — Checking that each candidate divides by 3 with a whole-number quotient.

💡 This only needs Grade 3 multiplication facts: list the multiples of 3 and see which land in the 50s!

Variant 5 answer: 0, 2, 4, 6, or 8

The division below comes out exactly (no remainder). Find every digit that $\blacksquare$ could be. (Here $9\blacksquare$ is a two-digit number.)

$9\blacksquare \div 2$

Show solution

Understand

The two-digit number 9-followed-by-a-blank must divide exactly by 2. We need every digit the blank could be so that the number is a multiple of 2.

Givens

The number is two digits and starts with 9 (it is 90 through 99).
When divided by 2 it must leave no remainder.

Unknowns

All possible digits for the blank (the ones digit).

Constraints

The blank is a single digit 0 through 9.
The whole number must be divisible by 2.

Plan

#2 Make a Systematic List · also uses: #6 Guess and Check

There are only ten numbers from 90 to 99, so we can list the multiples of 2 in that range and read off which ones-digits work.

Execute

#2 Make a Systematic List 3.OA.C.7

The multiples of 2 that land in the 90s are 90, 92, 94, 96, and 98. Any multiple outside 90 to 99 does not have 9 as its tens digit.

2 \times 45 = 90,\quad 2 \times 46 = 92,\quad 2 \times 47 = 94,\quad 2 \times 48 = 96,\quad 2 \times 49 = 98

Listing the multiples of 2 inside 90 to 99 catches every number that divides evenly.

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

Each of 90, 92, 94, 96, and 98 starts with 9, so the blank can be 0, 2, 4, 6, or 8. The other ones digits leave remainders, so they do not work.

90 \div 2 = 45,\quad 92 \div 2 = 46,\quad 94 \div 2 = 47,\quad 96 \div 2 = 48,\quad 98 \div 2 = 49

Each valid number gives a whole quotient, confirming the digit makes a clean division.

Answer: 0, 2, 4, 6, or 8

Review

90, 92, 94, 96, and 98 are the only multiples of 2 between 90 and 99 (multiples of 2 are spaced 2 apart, so a ten-wide window holds just a few). Each gives a whole quotient, so the answers are correct.

Test the ten numbers 90 through 99 one at a time, dividing each by 2, and keep the ones with no remainder.

Standards · min grade 3

3.OA.C.7 Fluently multiply and divide within 100 — Listing the multiples of 2 within the 90s.
3.OA.B.6 Understand division as an unknown-factor problem — Checking that each candidate divides by 2 with a whole-number quotient.

💡 This only needs Grade 3 multiplication facts: list the multiples of 2 and see which land in the 90s!

Variant 6 answer: 0 or 5

The division below comes out exactly (no remainder). Find every digit that $\blacksquare$ could be. (Here $3\blacksquare$ is a two-digit number.)

$3\blacksquare \div 5$

Show solution

Understand

The two-digit number 3-followed-by-a-blank must divide exactly by 5. We need every digit the blank could be so that the number is a multiple of 5.

Givens

The number is two digits and starts with 3 (it is 30 through 39).
When divided by 5 it must leave no remainder.

Unknowns

All possible digits for the blank (the ones digit).

Constraints

The blank is a single digit 0 through 9.
The whole number must be divisible by 5.

Plan

#2 Make a Systematic List · also uses: #6 Guess and Check

There are only ten numbers from 30 to 39, so we can list the multiples of 5 in that range and read off which ones-digits work.

Execute

#2 Make a Systematic List 3.OA.C.7

The multiples of 5 that land in the 30s are 30 and 35. Any multiple outside 30 to 39 does not have 3 as its tens digit.

5 \times 6 = 30,\quad 5 \times 7 = 35

Listing the multiples of 5 inside 30 to 39 catches every number that divides evenly.

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

Each of 30 and 35 starts with 3, so the blank can be 0 or 5. The other ones digits leave remainders, so they do not work.

30 \div 5 = 6,\quad 35 \div 5 = 7

Each valid number gives a whole quotient, confirming the digit makes a clean division.

Answer: 0 or 5

Review

30 and 35 are the only multiples of 5 between 30 and 39 (multiples of 5 are spaced 5 apart, so a ten-wide window holds just a few). Each gives a whole quotient, so the answers are correct.

Test the ten numbers 30 through 39 one at a time, dividing each by 5, and keep the ones with no remainder.

Standards · min grade 3

3.OA.C.7 Fluently multiply and divide within 100 — Listing the multiples of 5 within the 30s.
3.OA.B.6 Understand division as an unknown-factor problem — Checking that each candidate divides by 5 with a whole-number quotient.

💡 This only needs Grade 3 multiplication facts: list the multiples of 5 and see which land in the 30s!

Variant 7 answer: 1, 4, or 7

The division below comes out exactly (no remainder). Find every digit that $\blacksquare$ could be. (Here $2\blacksquare$ is a two-digit number.)

$2\blacksquare \div 3$

Show solution

Understand

The two-digit number 2-followed-by-a-blank must divide exactly by 3. We need every digit the blank could be so that the number is a multiple of 3.

Givens

The number is two digits and starts with 2 (it is 20 through 29).
When divided by 3 it must leave no remainder.

Unknowns

All possible digits for the blank (the ones digit).

Constraints

The blank is a single digit 0 through 9.
The whole number must be divisible by 3.

Plan

#2 Make a Systematic List · also uses: #6 Guess and Check

There are only ten numbers from 20 to 29, so we can list the multiples of 3 in that range and read off which ones-digits work.

Execute

#2 Make a Systematic List 3.OA.C.7

The multiples of 3 that land in the 20s are 21, 24, and 27. Any multiple outside 20 to 29 does not have 2 as its tens digit.

3 \times 7 = 21,\quad 3 \times 8 = 24,\quad 3 \times 9 = 27

Listing the multiples of 3 inside 20 to 29 catches every number that divides evenly.

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

Each of 21, 24, and 27 starts with 2, so the blank can be 1, 4, or 7. The other ones digits leave remainders, so they do not work.

21 \div 3 = 7,\quad 24 \div 3 = 8,\quad 27 \div 3 = 9

Each valid number gives a whole quotient, confirming the digit makes a clean division.

Answer: 1, 4, or 7

Review

21, 24, and 27 are the only multiples of 3 between 20 and 29 (multiples of 3 are spaced 3 apart, so a ten-wide window holds just a few). Each gives a whole quotient, so the answers are correct.

Test the ten numbers 20 through 29 one at a time, dividing each by 3, and keep the ones with no remainder.

Standards · min grade 3

3.OA.C.7 Fluently multiply and divide within 100 — Listing the multiples of 3 within the 20s.
3.OA.B.6 Understand division as an unknown-factor problem — Checking that each candidate divides by 3 with a whole-number quotient.

💡 This only needs Grade 3 multiplication facts: list the multiples of 3 and see which land in the 20s!

Variant 8 answer: 2 or 6

The division below comes out exactly (no remainder). Find every digit that $\blacksquare$ could be. (Here $1\blacksquare$ is a two-digit number.)

$1\blacksquare \div 4$

Show solution

Understand

The two-digit number 1-followed-by-a-blank must divide exactly by 4. We need every digit the blank could be so that the number is a multiple of 4.

Givens

The number is two digits and starts with 1 (it is 10 through 19).
When divided by 4 it must leave no remainder.

Unknowns

All possible digits for the blank (the ones digit).

Constraints

The blank is a single digit 0 through 9.
The whole number must be divisible by 4.

Plan

#2 Make a Systematic List · also uses: #6 Guess and Check

There are only ten numbers from 10 to 19, so we can list the multiples of 4 in that range and read off which ones-digits work.

Execute

#2 Make a Systematic List 3.OA.C.7

The multiples of 4 that land in the 10s are 12 and 16. Any multiple outside 10 to 19 does not have 1 as its tens digit.

4 \times 3 = 12,\quad 4 \times 4 = 16

Listing the multiples of 4 inside 10 to 19 catches every number that divides evenly.

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

Each of 12 and 16 starts with 1, so the blank can be 2 or 6. The other ones digits leave remainders, so they do not work.

12 \div 4 = 3,\quad 16 \div 4 = 4

Each valid number gives a whole quotient, confirming the digit makes a clean division.

Answer: 2 or 6

Review

12 and 16 are the only multiples of 4 between 10 and 19 (multiples of 4 are spaced 4 apart, so a ten-wide window holds just a few). Each gives a whole quotient, so the answers are correct.

Test the ten numbers 10 through 19 one at a time, dividing each by 4, and keep the ones with no remainder.

Standards · min grade 3

3.OA.C.7 Fluently multiply and divide within 100 — Listing the multiples of 4 within the 10s.
3.OA.B.6 Understand division as an unknown-factor problem — Checking that each candidate divides by 4 with a whole-number quotient.

💡 This only needs Grade 3 multiplication facts: list the multiples of 4 and see which land in the 10s!

Variant 9 answer: 0 or 5

The division below comes out exactly (no remainder). Find every digit that $\blacksquare$ could be. (Here $8\blacksquare$ is a two-digit number.)

$8\blacksquare \div 5$

Show solution

Understand

The two-digit number 8-followed-by-a-blank must divide exactly by 5. We need every digit the blank could be so that the number is a multiple of 5.

Givens

The number is two digits and starts with 8 (it is 80 through 89).
When divided by 5 it must leave no remainder.

Unknowns

All possible digits for the blank (the ones digit).

Constraints

The blank is a single digit 0 through 9.
The whole number must be divisible by 5.

Plan

#2 Make a Systematic List · also uses: #6 Guess and Check

There are only ten numbers from 80 to 89, so we can list the multiples of 5 in that range and read off which ones-digits work.

Execute

#2 Make a Systematic List 3.OA.C.7

The multiples of 5 that land in the 80s are 80 and 85. Any multiple outside 80 to 89 does not have 8 as its tens digit.

5 \times 16 = 80,\quad 5 \times 17 = 85

Listing the multiples of 5 inside 80 to 89 catches every number that divides evenly.

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

Each of 80 and 85 starts with 8, so the blank can be 0 or 5. The other ones digits leave remainders, so they do not work.

80 \div 5 = 16,\quad 85 \div 5 = 17

Each valid number gives a whole quotient, confirming the digit makes a clean division.

Answer: 0 or 5

Review

80 and 85 are the only multiples of 5 between 80 and 89 (multiples of 5 are spaced 5 apart, so a ten-wide window holds just a few). Each gives a whole quotient, so the answers are correct.

Test the ten numbers 80 through 89 one at a time, dividing each by 5, and keep the ones with no remainder.

Standards · min grade 3

3.OA.C.7 Fluently multiply and divide within 100 — Listing the multiples of 5 within the 80s.
3.OA.B.6 Understand division as an unknown-factor problem — Checking that each candidate divides by 5 with a whole-number quotient.

💡 This only needs Grade 3 multiplication facts: list the multiples of 5 and see which land in the 80s!

Variant 10 answer: 2 or 9

The division below comes out exactly (no remainder). Find every digit that $\blacksquare$ could be. (Here $4\blacksquare$ is a two-digit number.)

$4\blacksquare \div 7$

Show solution

Understand

The two-digit number 4-followed-by-a-blank must divide exactly by 7. We need every digit the blank could be so that the number is a multiple of 7.

Givens

The number is two digits and starts with 4 (it is 40 through 49).
When divided by 7 it must leave no remainder.

Unknowns

All possible digits for the blank (the ones digit).

Constraints

The blank is a single digit 0 through 9.
The whole number must be divisible by 7.

Plan

#2 Make a Systematic List · also uses: #6 Guess and Check

There are only ten numbers from 40 to 49, so we can list the multiples of 7 in that range and read off which ones-digits work.

Execute

#2 Make a Systematic List 3.OA.C.7

The multiples of 7 that land in the 40s are 42 and 49. Any multiple outside 40 to 49 does not have 4 as its tens digit.

7 \times 6 = 42,\quad 7 \times 7 = 49

Listing the multiples of 7 inside 40 to 49 catches every number that divides evenly.

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

Each of 42 and 49 starts with 4, so the blank can be 2 or 9. The other ones digits leave remainders, so they do not work.

42 \div 7 = 6,\quad 49 \div 7 = 7

Each valid number gives a whole quotient, confirming the digit makes a clean division.

Answer: 2 or 9

Review

42 and 49 are the only multiples of 7 between 40 and 49 (multiples of 7 are spaced 7 apart, so a ten-wide window holds just a few). Each gives a whole quotient, so the answers are correct.

Test the ten numbers 40 through 49 one at a time, dividing each by 7, and keep the ones with no remainder.

Standards · min grade 3

3.OA.C.7 Fluently multiply and divide within 100 — Listing the multiples of 7 within the 40s.
3.OA.B.6 Understand division as an unknown-factor problem — Checking that each candidate divides by 7 with a whole-number quotient.

💡 This only needs Grade 3 multiplication facts: list the multiples of 7 and see which land in the 40s!

Variant 11 answer: 2 or 6

The division below comes out exactly (no remainder). Find every digit that $\blacksquare$ could be. (Here $7\blacksquare$ is a two-digit number.)

$7\blacksquare \div 4$

Show solution

Understand

The two-digit number 7-followed-by-a-blank must divide exactly by 4. We need every digit the blank could be so that the number is a multiple of 4.

Givens

The number is two digits and starts with 7 (it is 70 through 79).
When divided by 4 it must leave no remainder.

Unknowns

All possible digits for the blank (the ones digit).

Constraints

The blank is a single digit 0 through 9.
The whole number must be divisible by 4.

Plan

#2 Make a Systematic List · also uses: #6 Guess and Check

There are only ten numbers from 70 to 79, so we can list the multiples of 4 in that range and read off which ones-digits work.

Execute

#2 Make a Systematic List 3.OA.C.7

The multiples of 4 that land in the 70s are 72 and 76. Any multiple outside 70 to 79 does not have 7 as its tens digit.

4 \times 18 = 72,\quad 4 \times 19 = 76

Listing the multiples of 4 inside 70 to 79 catches every number that divides evenly.

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

Each of 72 and 76 starts with 7, so the blank can be 2 or 6. The other ones digits leave remainders, so they do not work.

72 \div 4 = 18,\quad 76 \div 4 = 19

Each valid number gives a whole quotient, confirming the digit makes a clean division.

Answer: 2 or 6

Review

72 and 76 are the only multiples of 4 between 70 and 79 (multiples of 4 are spaced 4 apart, so a ten-wide window holds just a few). Each gives a whole quotient, so the answers are correct.

Test the ten numbers 70 through 79 one at a time, dividing each by 4, and keep the ones with no remainder.

Standards · min grade 3

3.OA.C.7 Fluently multiply and divide within 100 — Listing the multiples of 4 within the 70s.
3.OA.B.6 Understand division as an unknown-factor problem — Checking that each candidate divides by 4 with a whole-number quotient.

💡 This only needs Grade 3 multiplication facts: list the multiples of 4 and see which land in the 70s!