← 3-1 · Narrow candidates by divisibility conditions · Divisibility and Remainder Reasoning

Narrow candidates by divisibility conditions · 8 practice problems

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

Generated variants — 8

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

Variant 1 answer: 14

Find the two-digit number that satisfies all of the conditions below.

It is divisible by both $2$ and $7$ .
It is less than $60$ .
The sum of its tens digit and its ones digit is $5$ .

Show solution

Understand

We need a two-digit number that is divisible by both 2 and 7, is less than 60, and whose tens digit and ones digit add up to 5.

Givens

The number is divisible by 2 and also by 7.
The number is less than 60.
The tens digit plus the ones digit equals 5.
The number has two digits.

Unknowns

The two-digit number meeting all three conditions.

Constraints

Two-digit means it is between 10 and 99, but here also under 60.
Divisible by both 2 and 7 means divisible by 14.

Plan

#2 Make a Systematic List · also uses: #3 Eliminate Possibilities

Listing the numbers that satisfy the strongest clue (divisible by both 2 and 7, i.e. by 14) gives a tiny set, and then we eliminate any that fail the size or digit-sum clues.

Execute

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

A number divisible by both 2 and 7 is a multiple of 14. List the multiples of 14: 14, 28, 42, ... .

\text{multiples of }14: 14, 28, 42, \ldots

Grade 3 times-tables: being in both the 2 and 7 tables means being in the 14 table.

#3 Eliminate Possibilities 3.OA.B.6

Keep only the two-digit multiples of 14 under 60: that leaves 14, 28, 42, 56.

\text{candidates under }60: 14, 28, 42, 56

Grade 3: only the multiples of 14 that fit under 60 stay in the running.

#3 Eliminate Possibilities 3.OA.A.3

Check the digit sums of the candidates; only 14 has tens plus ones equal to 5 (1 + 4 = 5).

1 + 4 = 5

Grade 3 place value and addition: adding the two digits of 14 gives exactly 5.

Answer: 14

Review

Verify 14: 14 div 2 = 7 and 14 div 7 = 2 (divisible by both), 14 < 60 (true), and 1 + 4 = 5 (digit sum correct). All three conditions hold, and it is the only candidate that does.

List two-digit numbers under 60 whose digits add to 5, then keep only the one divisible by both 2 and 7, which is 14.

Standards · min grade 3

3.OA.C.7 Fluently multiply and divide within 100 — Listing multiples of 14 from the 2 and 7 times tables.
3.OA.B.6 Understand division as an unknown-factor problem — Confirming divisibility by 2 and 7 for the candidates.
3.OA.A.3 Solve multiplication and division word problems within 100 — Adding the digits to test the digit-sum condition.

💡 Divisible by 2 and 7 means a multiple of 14, and the only one under 60 with digits adding to 5 is 14!

Variant 2 answer: 15

Find the two-digit number that satisfies all of the conditions below.

It is divisible by both $3$ and $5$ .
It is less than $50$ .
The sum of its tens digit and its ones digit is $6$ .

Show solution

Understand

We need a two-digit number that is divisible by both 3 and 5, is less than 50, and whose tens digit and ones digit add up to 6.

Givens

The number is divisible by 3 and also by 5.
The number is less than 50.
The tens digit plus the ones digit equals 6.
The number has two digits.

Unknowns

The two-digit number meeting all three conditions.

Constraints

Two-digit means it is between 10 and 99, but here also under 50.
Divisible by both 3 and 5 means divisible by 15.

Plan

#2 Make a Systematic List · also uses: #3 Eliminate Possibilities

Listing the numbers that satisfy the strongest clue (divisible by both 3 and 5, i.e. by 15) gives a tiny set, and then we eliminate any that fail the size or digit-sum clues.

Execute

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

A number divisible by both 3 and 5 is a multiple of 15. List the multiples of 15: 15, 30, 45, ... .

\text{multiples of }15: 15, 30, 45, \ldots

Grade 3 times-tables: being in both the 3 and 5 tables means being in the 15 table.

#3 Eliminate Possibilities 3.OA.B.6

Keep only the two-digit multiples of 15 under 50: that leaves 15, 30, 45.

\text{candidates under }50: 15, 30, 45

Grade 3: only the multiples of 15 that fit under 50 stay in the running.

#3 Eliminate Possibilities 3.OA.A.3

Check the digit sums of the candidates; only 15 has tens plus ones equal to 6 (1 + 5 = 6).

1 + 5 = 6

Grade 3 place value and addition: adding the two digits of 15 gives exactly 6.

Answer: 15

Review

Verify 15: 15 div 3 = 5 and 15 div 5 = 3 (divisible by both), 15 < 50 (true), and 1 + 5 = 6 (digit sum correct). All three conditions hold, and it is the only candidate that does.

List two-digit numbers under 50 whose digits add to 6, then keep only the one divisible by both 3 and 5, which is 15.

Standards · min grade 3

3.OA.C.7 Fluently multiply and divide within 100 — Listing multiples of 15 from the 3 and 5 times tables.
3.OA.B.6 Understand division as an unknown-factor problem — Confirming divisibility by 3 and 5 for the candidates.
3.OA.A.3 Solve multiplication and division word problems within 100 — Adding the digits to test the digit-sum condition.

💡 Divisible by 3 and 5 means a multiple of 15, and the only one under 50 with digits adding to 6 is 15!

Variant 3 answer: 24

Find the two-digit number that satisfies all of the conditions below.

It is divisible by both $4$ and $6$ .
It is less than $30$ .
The sum of its tens digit and its ones digit is $6$ .

Show solution

Understand

We need a two-digit number that is divisible by both 4 and 6, is less than 30, and whose tens digit and ones digit add up to 6.

Givens

The number is divisible by 4 and also by 6.
The number is less than 30.
The tens digit plus the ones digit equals 6.
The number has two digits.

Unknowns

The two-digit number meeting all three conditions.

Constraints

Two-digit means it is between 10 and 99, but here also under 30.
Divisible by both 4 and 6 means divisible by 12.

Plan

#2 Make a Systematic List · also uses: #3 Eliminate Possibilities

Listing the numbers that satisfy the strongest clue (divisible by both 4 and 6, i.e. by 12) gives a tiny set, and then we eliminate any that fail the size or digit-sum clues.

Execute

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

A number divisible by both 4 and 6 is a multiple of 12. List the multiples of 12: 12, 24, 36, ... .

\text{multiples of }12: 12, 24, 36, \ldots

Grade 3 times-tables: being in both the 4 and 6 tables means being in the 12 table.

#3 Eliminate Possibilities 3.OA.B.6

Keep only the two-digit multiples of 12 under 30: that leaves 12, 24.

\text{candidates under }30: 12, 24

Grade 3: only the multiples of 12 that fit under 30 stay in the running.

#3 Eliminate Possibilities 3.OA.A.3

Check the digit sums of the candidates; only 24 has tens plus ones equal to 6 (2 + 4 = 6).

2 + 4 = 6

Grade 3 place value and addition: adding the two digits of 24 gives exactly 6.

Answer: 24

Review

Verify 24: 24 div 4 = 6 and 24 div 6 = 4 (divisible by both), 24 < 30 (true), and 2 + 4 = 6 (digit sum correct). All three conditions hold, and it is the only candidate that does.

List two-digit numbers under 30 whose digits add to 6, then keep only the one divisible by both 4 and 6, which is 24.

Standards · min grade 3

3.OA.C.7 Fluently multiply and divide within 100 — Listing multiples of 12 from the 4 and 6 times tables.
3.OA.B.6 Understand division as an unknown-factor problem — Confirming divisibility by 4 and 6 for the candidates.
3.OA.A.3 Solve multiplication and division word problems within 100 — Adding the digits to test the digit-sum condition.

💡 Divisible by 4 and 6 means a multiple of 12, and the only one under 30 with digits adding to 6 is 24!

Variant 4 answer: 24

Find the two-digit number that satisfies all of the conditions below.

It is divisible by both $3$ and $8$ .
It is less than $80$ .
The sum of its tens digit and its ones digit is $6$ .

Show solution

Understand

We need a two-digit number that is divisible by both 3 and 8, is less than 80, and whose tens digit and ones digit add up to 6.

Givens

The number is divisible by 3 and also by 8.
The number is less than 80.
The tens digit plus the ones digit equals 6.
The number has two digits.

Unknowns

The two-digit number meeting all three conditions.

Constraints

Two-digit means it is between 10 and 99, but here also under 80.
Divisible by both 3 and 8 means divisible by 24.

Plan

#2 Make a Systematic List · also uses: #3 Eliminate Possibilities

Listing the numbers that satisfy the strongest clue (divisible by both 3 and 8, i.e. by 24) gives a tiny set, and then we eliminate any that fail the size or digit-sum clues.

Execute

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

A number divisible by both 3 and 8 is a multiple of 24. List the multiples of 24: 24, 48, 72, ... .

\text{multiples of }24: 24, 48, 72, \ldots

Grade 3 times-tables: being in both the 3 and 8 tables means being in the 24 table.

#3 Eliminate Possibilities 3.OA.B.6

Keep only the two-digit multiples of 24 under 80: that leaves 24, 48, 72.

\text{candidates under }80: 24, 48, 72

Grade 3: only the multiples of 24 that fit under 80 stay in the running.

#3 Eliminate Possibilities 3.OA.A.3

Check the digit sums of the candidates; only 24 has tens plus ones equal to 6 (2 + 4 = 6).

2 + 4 = 6

Grade 3 place value and addition: adding the two digits of 24 gives exactly 6.

Answer: 24

Review

Verify 24: 24 div 3 = 8 and 24 div 8 = 3 (divisible by both), 24 < 80 (true), and 2 + 4 = 6 (digit sum correct). All three conditions hold, and it is the only candidate that does.

List two-digit numbers under 80 whose digits add to 6, then keep only the one divisible by both 3 and 8, which is 24.

Standards · min grade 3

3.OA.C.7 Fluently multiply and divide within 100 — Listing multiples of 24 from the 3 and 8 times tables.
3.OA.B.6 Understand division as an unknown-factor problem — Confirming divisibility by 3 and 8 for the candidates.
3.OA.A.3 Solve multiplication and division word problems within 100 — Adding the digits to test the digit-sum condition.

💡 Divisible by 3 and 8 means a multiple of 24, and the only one under 80 with digits adding to 6 is 24!

Variant 5 answer: 70

Find the two-digit number that satisfies all of the conditions below.

It is divisible by both $2$ and $5$ .
It is less than $80$ .
The sum of its tens digit and its ones digit is $7$ .

Show solution

Understand

We need a two-digit number that is divisible by both 2 and 5, is less than 80, and whose tens digit and ones digit add up to 7.

Givens

The number is divisible by 2 and also by 5.
The number is less than 80.
The tens digit plus the ones digit equals 7.
The number has two digits.

Unknowns

The two-digit number meeting all three conditions.

Constraints

Two-digit means it is between 10 and 99, but here also under 80.
Divisible by both 2 and 5 means divisible by 10.

Plan

#2 Make a Systematic List · also uses: #3 Eliminate Possibilities

Listing the numbers that satisfy the strongest clue (divisible by both 2 and 5, i.e. by 10) gives a tiny set, and then we eliminate any that fail the size or digit-sum clues.

Execute

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

A number divisible by both 2 and 5 is a multiple of 10. List the multiples of 10: 10, 20, 30, ... .

\text{multiples of }10: 10, 20, 30, \ldots

Grade 3 times-tables: being in both the 2 and 5 tables means being in the 10 table.

#3 Eliminate Possibilities 3.OA.B.6

Keep only the two-digit multiples of 10 under 80: that leaves 10, 20, 30, 40, 50, 60, 70.

\text{candidates under }80: 10, 20, 30, 40, 50, 60, 70

Grade 3: only the multiples of 10 that fit under 80 stay in the running.

#3 Eliminate Possibilities 3.OA.A.3

Check the digit sums of the candidates; only 70 has tens plus ones equal to 7 (7 + 0 = 7).

7 + 0 = 7

Grade 3 place value and addition: adding the two digits of 70 gives exactly 7.

Answer: 70

Review

Verify 70: 70 div 2 = 35 and 70 div 5 = 14 (divisible by both), 70 < 80 (true), and 7 + 0 = 7 (digit sum correct). All three conditions hold, and it is the only candidate that does.

List two-digit numbers under 80 whose digits add to 7, then keep only the one divisible by both 2 and 5, which is 70.

Standards · min grade 3

3.OA.C.7 Fluently multiply and divide within 100 — Listing multiples of 10 from the 2 and 5 times tables.
3.OA.B.6 Understand division as an unknown-factor problem — Confirming divisibility by 2 and 5 for the candidates.
3.OA.A.3 Solve multiplication and division word problems within 100 — Adding the digits to test the digit-sum condition.

💡 Divisible by 2 and 5 means a multiple of 10, and the only one under 80 with digits adding to 7 is 70!

Variant 6 answer: 40

Find the two-digit number that satisfies all of the conditions below.

It is divisible by both $4$ and $5$ .
It is less than $90$ .
The sum of its tens digit and its ones digit is $4$ .

Show solution

Understand

We need a two-digit number that is divisible by both 4 and 5, is less than 90, and whose tens digit and ones digit add up to 4.

Givens

The number is divisible by 4 and also by 5.
The number is less than 90.
The tens digit plus the ones digit equals 4.
The number has two digits.

Unknowns

The two-digit number meeting all three conditions.

Constraints

Two-digit means it is between 10 and 99, but here also under 90.
Divisible by both 4 and 5 means divisible by 20.

Plan

#2 Make a Systematic List · also uses: #3 Eliminate Possibilities

Listing the numbers that satisfy the strongest clue (divisible by both 4 and 5, i.e. by 20) gives a tiny set, and then we eliminate any that fail the size or digit-sum clues.

Execute

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

A number divisible by both 4 and 5 is a multiple of 20. List the multiples of 20: 20, 40, 60, ... .

\text{multiples of }20: 20, 40, 60, \ldots

Grade 3 times-tables: being in both the 4 and 5 tables means being in the 20 table.

#3 Eliminate Possibilities 3.OA.B.6

Keep only the two-digit multiples of 20 under 90: that leaves 20, 40, 60, 80.

\text{candidates under }90: 20, 40, 60, 80

Grade 3: only the multiples of 20 that fit under 90 stay in the running.

#3 Eliminate Possibilities 3.OA.A.3

Check the digit sums of the candidates; only 40 has tens plus ones equal to 4 (4 + 0 = 4).

4 + 0 = 4

Grade 3 place value and addition: adding the two digits of 40 gives exactly 4.

Answer: 40

Review

Verify 40: 40 div 4 = 10 and 40 div 5 = 8 (divisible by both), 40 < 90 (true), and 4 + 0 = 4 (digit sum correct). All three conditions hold, and it is the only candidate that does.

List two-digit numbers under 90 whose digits add to 4, then keep only the one divisible by both 4 and 5, which is 40.

Standards · min grade 3

3.OA.C.7 Fluently multiply and divide within 100 — Listing multiples of 20 from the 4 and 5 times tables.
3.OA.B.6 Understand division as an unknown-factor problem — Confirming divisibility by 4 and 5 for the candidates.
3.OA.A.3 Solve multiplication and division word problems within 100 — Adding the digits to test the digit-sum condition.

💡 Divisible by 4 and 5 means a multiple of 20, and the only one under 90 with digits adding to 4 is 40!

Variant 7 answer: 12

Find the two-digit number that satisfies all of the conditions below.

It is divisible by both $4$ and $6$ .
It is less than $50$ .
The sum of its tens digit and its ones digit is $3$ .

Show solution

Understand

We need a two-digit number that is divisible by both 4 and 6, is less than 50, and whose tens digit and ones digit add up to 3.

Givens

The number is divisible by 4 and also by 6.
The number is less than 50.
The tens digit plus the ones digit equals 3.
The number has two digits.

Unknowns

The two-digit number meeting all three conditions.

Constraints

Two-digit means it is between 10 and 99, but here also under 50.
Divisible by both 4 and 6 means divisible by 12.

Plan

#2 Make a Systematic List · also uses: #3 Eliminate Possibilities

Listing the numbers that satisfy the strongest clue (divisible by both 4 and 6, i.e. by 12) gives a tiny set, and then we eliminate any that fail the size or digit-sum clues.

Execute

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

A number divisible by both 4 and 6 is a multiple of 12. List the multiples of 12: 12, 24, 36, ... .

\text{multiples of }12: 12, 24, 36, \ldots

Grade 3 times-tables: being in both the 4 and 6 tables means being in the 12 table.

#3 Eliminate Possibilities 3.OA.B.6

Keep only the two-digit multiples of 12 under 50: that leaves 12, 24, 36, 48.

\text{candidates under }50: 12, 24, 36, 48

Grade 3: only the multiples of 12 that fit under 50 stay in the running.

#3 Eliminate Possibilities 3.OA.A.3

Check the digit sums of the candidates; only 12 has tens plus ones equal to 3 (1 + 2 = 3).

1 + 2 = 3

Grade 3 place value and addition: adding the two digits of 12 gives exactly 3.

Answer: 12

Review

Verify 12: 12 div 4 = 3 and 12 div 6 = 2 (divisible by both), 12 < 50 (true), and 1 + 2 = 3 (digit sum correct). All three conditions hold, and it is the only candidate that does.

List two-digit numbers under 50 whose digits add to 3, then keep only the one divisible by both 4 and 6, which is 12.

Standards · min grade 3

3.OA.C.7 Fluently multiply and divide within 100 — Listing multiples of 12 from the 4 and 6 times tables.
3.OA.B.6 Understand division as an unknown-factor problem — Confirming divisibility by 4 and 6 for the candidates.
3.OA.A.3 Solve multiplication and division word problems within 100 — Adding the digits to test the digit-sum condition.

💡 Divisible by 4 and 6 means a multiple of 12, and the only one under 50 with digits adding to 3 is 12!

Variant 8 answer: 24

Find the two-digit number that satisfies all of the conditions below.

It is divisible by both $3$ and $4$ .
It is less than $50$ .
The sum of its tens digit and its ones digit is $6$ .

Show solution

Understand

We need a two-digit number that is divisible by both 3 and 4, is less than 50, and whose tens digit and ones digit add up to 6.

Givens

The number is divisible by 3 and also by 4.
The number is less than 50.
The tens digit plus the ones digit equals 6.
The number has two digits.

Unknowns

The two-digit number meeting all three conditions.

Constraints

Two-digit means it is between 10 and 99, but here also under 50.
Divisible by both 3 and 4 means divisible by 12.

Plan

#2 Make a Systematic List · also uses: #3 Eliminate Possibilities

Listing the numbers that satisfy the strongest clue (divisible by both 3 and 4, i.e. by 12) gives a tiny set, and then we eliminate any that fail the size or digit-sum clues.

Execute

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

A number divisible by both 3 and 4 is a multiple of 12. List the multiples of 12: 12, 24, 36, ... .

\text{multiples of }12: 12, 24, 36, \ldots

Grade 3 times-tables: being in both the 3 and 4 tables means being in the 12 table.

#3 Eliminate Possibilities 3.OA.B.6

Keep only the two-digit multiples of 12 under 50: that leaves 12, 24, 36, 48.

\text{candidates under }50: 12, 24, 36, 48

Grade 3: only the multiples of 12 that fit under 50 stay in the running.

#3 Eliminate Possibilities 3.OA.A.3

Check the digit sums of the candidates; only 24 has tens plus ones equal to 6 (2 + 4 = 6).

2 + 4 = 6

Grade 3 place value and addition: adding the two digits of 24 gives exactly 6.

Answer: 24

Review

Verify 24: 24 div 3 = 8 and 24 div 4 = 6 (divisible by both), 24 < 50 (true), and 2 + 4 = 6 (digit sum correct). All three conditions hold, and it is the only candidate that does.

List two-digit numbers under 50 whose digits add to 6, then keep only the one divisible by both 3 and 4, which is 24.

Standards · min grade 3

3.OA.C.7 Fluently multiply and divide within 100 — Listing multiples of 12 from the 3 and 4 times tables.
3.OA.B.6 Understand division as an unknown-factor problem — Confirming divisibility by 3 and 4 for the candidates.
3.OA.A.3 Solve multiplication and division word problems within 100 — Adding the digits to test the digit-sum condition.

💡 Divisible by 3 and 4 means a multiple of 12, and the only one under 50 with digits adding to 6 is 24!