← Count equal-share ways using divisors · Divisibility and Remainder Reasoning

Count equal-share ways using divisors · 12 practice problems

4.OA.B.43.OA.B.6

Generated variants — 12

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

Variant 1 easy answer: 4 ways

You want to share $24$ candies equally, with none left over, among more than $5$ students. In how many different ways can the candies be shared?

Show solution

1 · Understandwhat's really being asked

24 candies are split evenly among a group of more than 5 students, with nothing left over. We must count how many different group sizes make that possible.

Givens

There are 24 candies in total.
Everyone gets the same number, and none are left over.
The number of students is more than 5.

Unknowns

How many different group sizes work.

Constraints

Each student's share is a whole number of candies.

2 · Planchoose the strategy

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

'Equal shares with none left over' is the definition of a factor, so list the factors of 24 in pairs -- that way none is missed -- and then keep the ones big enough.

3 · Execute4 carry out the plan

1Turn 'equal shares, none left over' into divisors

#2 Make a Systematic List 3.OA.B.6

If 24 candies split evenly among some students, that number of students divides 24 exactly.

24 \div \square = \text{whole number}

The question is really about factors, not about candy.

2List all factor pairs of 24

#2 Make a Systematic List 4.OA.B.4

Pairing them up guarantees the list is complete: work upwards until the two halves of a pair meet.

24 = 1 \times 24, 2 \times 12, 3 \times 8, 4 \times 6

The factors are 1, 2, 3, 4, 6, 8, 12, 24.

3Keep only group sizes greater than 5

#6 Guess and Check 4.OA.B.4

The group must be bigger than 5, which rules out 1, 2, 3, 4.

6, 8, 12, 24

That leaves 4 group sizes.

4Confirm each surviving split works

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

Divide to see each one really does share out evenly.

24 \div 6 = 4,\quad 24 \div 8 = 3,\quad 24 \div 12 = 2,\quad 24 \div 24 = 1

All 4 work, so there are 4 ways.

Answer: 4 ways

4 · Reviewdoes it hold up?

24 has 8 factors in all, and 4 of them are 5 or less, leaving 4.

Another way: Test each number from 6 up to 24 and keep the ones that divide 24; slower, but it gives the same 4.

Standardsmin grade 4

4.OA.B.4 Find all factor pairs and recognize multiples; determine prime or composite — Listing every factor of 24 in pairs and filtering by size.
3.OA.B.6 Understand division as an unknown-factor problem — Reading 'shares evenly with none left over' as exact division.

💡Takeaway. 'Shares evenly with none left over' is another way of saying 'is a factor' -- so list the factor pairs and nothing is missed.

Variant 2 easy answer: 3 ways

You want to share $30$ candies equally, with none left over, among more than $6$ students. In how many different ways can the candies be shared?

Show solution

1 · Understandwhat's really being asked

30 candies are split evenly among a group of more than 6 students, with nothing left over. We must count how many different group sizes make that possible.

Givens

There are 30 candies in total.
Everyone gets the same number, and none are left over.
The number of students is more than 6.

Unknowns

How many different group sizes work.

Constraints

Each student's share is a whole number of candies.

2 · Planchoose the strategy

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

'Equal shares with none left over' is the definition of a factor, so list the factors of 30 in pairs -- that way none is missed -- and then keep the ones big enough.

3 · Execute4 carry out the plan

1Turn 'equal shares, none left over' into divisors

#2 Make a Systematic List 3.OA.B.6

If 30 candies split evenly among some students, that number of students divides 30 exactly.

30 \div \square = \text{whole number}

The question is really about factors, not about candy.

2List all factor pairs of 30

#2 Make a Systematic List 4.OA.B.4

Pairing them up guarantees the list is complete: work upwards until the two halves of a pair meet.

30 = 1 \times 30, 2 \times 15, 3 \times 10, 5 \times 6

The factors are 1, 2, 3, 5, 6, 10, 15, 30.

3Keep only group sizes greater than 6

#6 Guess and Check 4.OA.B.4

The group must be bigger than 6, which rules out 1, 2, 3, 5, 6.

10, 15, 30

That leaves 3 group sizes.

4Confirm each surviving split works

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

Divide to see each one really does share out evenly.

30 \div 10 = 3,\quad 30 \div 15 = 2,\quad 30 \div 30 = 1

All 3 work, so there are 3 ways.

Answer: 3 ways

4 · Reviewdoes it hold up?

30 has 8 factors in all, and 5 of them are 6 or less, leaving 3.

Another way: Test each number from 7 up to 30 and keep the ones that divide 30; slower, but it gives the same 3.

Standardsmin grade 4

4.OA.B.4 Find all factor pairs and recognize multiples; determine prime or composite — Listing every factor of 30 in pairs and filtering by size.
3.OA.B.6 Understand division as an unknown-factor problem — Reading 'shares evenly with none left over' as exact division.

💡Takeaway. 'Shares evenly with none left over' is another way of saying 'is a factor' -- so list the factor pairs and nothing is missed.

Variant 3 easy answer: 4 ways

You want to share $36$ candies equally, with none left over, among more than $8$ students. In how many different ways can the candies be shared?

Show solution

1 · Understandwhat's really being asked

36 candies are split evenly among a group of more than 8 students, with nothing left over. We must count how many different group sizes make that possible.

Givens

There are 36 candies in total.
Everyone gets the same number, and none are left over.
The number of students is more than 8.

Unknowns

How many different group sizes work.

Constraints

Each student's share is a whole number of candies.

2 · Planchoose the strategy

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

'Equal shares with none left over' is the definition of a factor, so list the factors of 36 in pairs -- that way none is missed -- and then keep the ones big enough.

3 · Execute4 carry out the plan

1Turn 'equal shares, none left over' into divisors

#2 Make a Systematic List 3.OA.B.6

If 36 candies split evenly among some students, that number of students divides 36 exactly.

36 \div \square = \text{whole number}

The question is really about factors, not about candy.

2List all factor pairs of 36

#2 Make a Systematic List 4.OA.B.4

Pairing them up guarantees the list is complete: work upwards until the two halves of a pair meet.

36 = 1 \times 36, 2 \times 18, 3 \times 12, 4 \times 9, 6 \times 6

The factors are 1, 2, 3, 4, 6, 9, 12, 18, 36.

3Keep only group sizes greater than 8

#6 Guess and Check 4.OA.B.4

The group must be bigger than 8, which rules out 1, 2, 3, 4, 6.

9, 12, 18, 36

That leaves 4 group sizes.

4Confirm each surviving split works

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

Divide to see each one really does share out evenly.

36 \div 9 = 4,\quad 36 \div 12 = 3,\quad 36 \div 18 = 2,\quad 36 \div 36 = 1

All 4 work, so there are 4 ways.

Answer: 4 ways

4 · Reviewdoes it hold up?

36 has 9 factors in all, and 5 of them are 8 or less, leaving 4.

Another way: Test each number from 9 up to 36 and keep the ones that divide 36; slower, but it gives the same 4.

Standardsmin grade 4

4.OA.B.4 Find all factor pairs and recognize multiples; determine prime or composite — Listing every factor of 36 in pairs and filtering by size.
3.OA.B.6 Understand division as an unknown-factor problem — Reading 'shares evenly with none left over' as exact division.

💡Takeaway. 'Shares evenly with none left over' is another way of saying 'is a factor' -- so list the factor pairs and nothing is missed.

Variant 4 easy answer: 3 ways

You want to share $42$ candies equally, with none left over, among more than $10$ students. In how many different ways can the candies be shared?

Show solution

1 · Understandwhat's really being asked

42 candies are split evenly among a group of more than 10 students, with nothing left over. We must count how many different group sizes make that possible.

Givens

There are 42 candies in total.
Everyone gets the same number, and none are left over.
The number of students is more than 10.

Unknowns

How many different group sizes work.

Constraints

Each student's share is a whole number of candies.

2 · Planchoose the strategy

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

'Equal shares with none left over' is the definition of a factor, so list the factors of 42 in pairs -- that way none is missed -- and then keep the ones big enough.

3 · Execute4 carry out the plan

1Turn 'equal shares, none left over' into divisors

#2 Make a Systematic List 3.OA.B.6

If 42 candies split evenly among some students, that number of students divides 42 exactly.

42 \div \square = \text{whole number}

The question is really about factors, not about candy.

2List all factor pairs of 42

#2 Make a Systematic List 4.OA.B.4

Pairing them up guarantees the list is complete: work upwards until the two halves of a pair meet.

42 = 1 \times 42, 2 \times 21, 3 \times 14, 6 \times 7

The factors are 1, 2, 3, 6, 7, 14, 21, 42.

3Keep only group sizes greater than 10

#6 Guess and Check 4.OA.B.4

The group must be bigger than 10, which rules out 1, 2, 3, 6, 7.

14, 21, 42

That leaves 3 group sizes.

4Confirm each surviving split works

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

Divide to see each one really does share out evenly.

42 \div 14 = 3,\quad 42 \div 21 = 2,\quad 42 \div 42 = 1

All 3 work, so there are 3 ways.

Answer: 3 ways

4 · Reviewdoes it hold up?

42 has 8 factors in all, and 5 of them are 10 or less, leaving 3.

Another way: Test each number from 11 up to 42 and keep the ones that divide 42; slower, but it gives the same 3.

Standardsmin grade 4

4.OA.B.4 Find all factor pairs and recognize multiples; determine prime or composite — Listing every factor of 42 in pairs and filtering by size.
3.OA.B.6 Understand division as an unknown-factor problem — Reading 'shares evenly with none left over' as exact division.

💡Takeaway. 'Shares evenly with none left over' is another way of saying 'is a factor' -- so list the factor pairs and nothing is missed.

Variant 5 medium answer: 3 ways

You want to share $48$ candies equally, with none left over, among more than $15$ students. In how many different ways can the candies be shared?

Show solution

1 · Understandwhat's really being asked

48 candies are split evenly among a group of more than 15 students, with nothing left over. We must count how many different group sizes make that possible.

Givens

There are 48 candies in total.
Everyone gets the same number, and none are left over.
The number of students is more than 15.

Unknowns

How many different group sizes work.

Constraints

Each student's share is a whole number of candies.

2 · Planchoose the strategy

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

'Equal shares with none left over' is the definition of a factor, so list the factors of 48 in pairs -- that way none is missed -- and then keep the ones big enough.

3 · Execute4 carry out the plan

1Turn 'equal shares, none left over' into divisors

#2 Make a Systematic List 3.OA.B.6

If 48 candies split evenly among some students, that number of students divides 48 exactly.

48 \div \square = \text{whole number}

The question is really about factors, not about candy.

2List all factor pairs of 48

#2 Make a Systematic List 4.OA.B.4

Pairing them up guarantees the list is complete: work upwards until the two halves of a pair meet.

48 = 1 \times 48, 2 \times 24, 3 \times 16, 4 \times 12, 6 \times 8

The factors are 1, 2, 3, 4, 6, 8, 12, 16, 24, 48.

3Keep only group sizes greater than 15

#6 Guess and Check 4.OA.B.4

The group must be bigger than 15, which rules out 1, 2, 3, 4, 6, 8, 12.

16, 24, 48

That leaves 3 group sizes.

4Confirm each surviving split works

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

Divide to see each one really does share out evenly.

48 \div 16 = 3,\quad 48 \div 24 = 2,\quad 48 \div 48 = 1

All 3 work, so there are 3 ways.

Answer: 3 ways

4 · Reviewdoes it hold up?

48 has 10 factors in all, and 7 of them are 15 or less, leaving 3.

Another way: Test each number from 16 up to 48 and keep the ones that divide 48; slower, but it gives the same 3.

Standardsmin grade 4

4.OA.B.4 Find all factor pairs and recognize multiples; determine prime or composite — Listing every factor of 48 in pairs and filtering by size.
3.OA.B.6 Understand division as an unknown-factor problem — Reading 'shares evenly with none left over' as exact division.

💡Takeaway. 'Shares evenly with none left over' is another way of saying 'is a factor' -- so list the factor pairs and nothing is missed.

Variant 6 medium answer: 3 ways

You want to share $54$ candies equally, with none left over, among more than $9$ students. In how many different ways can the candies be shared?

Show solution

1 · Understandwhat's really being asked

54 candies are split evenly among a group of more than 9 students, with nothing left over. We must count how many different group sizes make that possible.

Givens

There are 54 candies in total.
Everyone gets the same number, and none are left over.
The number of students is more than 9.

Unknowns

How many different group sizes work.

Constraints

Each student's share is a whole number of candies.

2 · Planchoose the strategy

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

'Equal shares with none left over' is the definition of a factor, so list the factors of 54 in pairs -- that way none is missed -- and then keep the ones big enough.

3 · Execute4 carry out the plan

1Turn 'equal shares, none left over' into divisors

#2 Make a Systematic List 3.OA.B.6

If 54 candies split evenly among some students, that number of students divides 54 exactly.

54 \div \square = \text{whole number}

The question is really about factors, not about candy.

2List all factor pairs of 54

#2 Make a Systematic List 4.OA.B.4

Pairing them up guarantees the list is complete: work upwards until the two halves of a pair meet.

54 = 1 \times 54, 2 \times 27, 3 \times 18, 6 \times 9

The factors are 1, 2, 3, 6, 9, 18, 27, 54.

3Keep only group sizes greater than 9

#6 Guess and Check 4.OA.B.4

The group must be bigger than 9, which rules out 1, 2, 3, 6, 9.

18, 27, 54

That leaves 3 group sizes.

4Confirm each surviving split works

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

Divide to see each one really does share out evenly.

54 \div 18 = 3,\quad 54 \div 27 = 2,\quad 54 \div 54 = 1

All 3 work, so there are 3 ways.

Answer: 3 ways

4 · Reviewdoes it hold up?

54 has 8 factors in all, and 5 of them are 9 or less, leaving 3.

Another way: Test each number from 10 up to 54 and keep the ones that divide 54; slower, but it gives the same 3.

Standardsmin grade 4

4.OA.B.4 Find all factor pairs and recognize multiples; determine prime or composite — Listing every factor of 54 in pairs and filtering by size.
3.OA.B.6 Understand division as an unknown-factor problem — Reading 'shares evenly with none left over' as exact division.

💡Takeaway. 'Shares evenly with none left over' is another way of saying 'is a factor' -- so list the factor pairs and nothing is missed.

Variant 7 medium answer: 4 ways

You want to share $60$ candies equally, with none left over, among more than $12$ students. In how many different ways can the candies be shared?

Show solution

1 · Understandwhat's really being asked

60 candies are split evenly among a group of more than 12 students, with nothing left over. We must count how many different group sizes make that possible.

Givens

There are 60 candies in total.
Everyone gets the same number, and none are left over.
The number of students is more than 12.

Unknowns

How many different group sizes work.

Constraints

Each student's share is a whole number of candies.

2 · Planchoose the strategy

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

'Equal shares with none left over' is the definition of a factor, so list the factors of 60 in pairs -- that way none is missed -- and then keep the ones big enough.

3 · Execute4 carry out the plan

1Turn 'equal shares, none left over' into divisors

#2 Make a Systematic List 3.OA.B.6

If 60 candies split evenly among some students, that number of students divides 60 exactly.

60 \div \square = \text{whole number}

The question is really about factors, not about candy.

2List all factor pairs of 60

#2 Make a Systematic List 4.OA.B.4

Pairing them up guarantees the list is complete: work upwards until the two halves of a pair meet.

60 = 1 \times 60, 2 \times 30, 3 \times 20, 4 \times 15, 5 \times 12, 6 \times 10

The factors are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60.

3Keep only group sizes greater than 12

#6 Guess and Check 4.OA.B.4

The group must be bigger than 12, which rules out 1, 2, 3, 4, 5, 6, 10, 12.

15, 20, 30, 60

That leaves 4 group sizes.

4Confirm each surviving split works

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

Divide to see each one really does share out evenly.

60 \div 15 = 4,\quad 60 \div 20 = 3,\quad 60 \div 30 = 2,\quad 60 \div 60 = 1

All 4 work, so there are 4 ways.

Answer: 4 ways

4 · Reviewdoes it hold up?

60 has 12 factors in all, and 8 of them are 12 or less, leaving 4.

Another way: Test each number from 13 up to 60 and keep the ones that divide 60; slower, but it gives the same 4.

Standardsmin grade 4

4.OA.B.4 Find all factor pairs and recognize multiples; determine prime or composite — Listing every factor of 60 in pairs and filtering by size.
3.OA.B.6 Understand division as an unknown-factor problem — Reading 'shares evenly with none left over' as exact division.

💡Takeaway. 'Shares evenly with none left over' is another way of saying 'is a factor' -- so list the factor pairs and nothing is missed.

Variant 8 medium answer: 3 ways

You want to share $66$ candies equally, with none left over, among more than $11$ students. In how many different ways can the candies be shared?

Show solution

1 · Understandwhat's really being asked

66 candies are split evenly among a group of more than 11 students, with nothing left over. We must count how many different group sizes make that possible.

Givens

There are 66 candies in total.
Everyone gets the same number, and none are left over.
The number of students is more than 11.

Unknowns

How many different group sizes work.

Constraints

Each student's share is a whole number of candies.

2 · Planchoose the strategy

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

'Equal shares with none left over' is the definition of a factor, so list the factors of 66 in pairs -- that way none is missed -- and then keep the ones big enough.

3 · Execute4 carry out the plan

1Turn 'equal shares, none left over' into divisors

#2 Make a Systematic List 3.OA.B.6

If 66 candies split evenly among some students, that number of students divides 66 exactly.

66 \div \square = \text{whole number}

The question is really about factors, not about candy.

2List all factor pairs of 66

#2 Make a Systematic List 4.OA.B.4

Pairing them up guarantees the list is complete: work upwards until the two halves of a pair meet.

66 = 1 \times 66, 2 \times 33, 3 \times 22, 6 \times 11

The factors are 1, 2, 3, 6, 11, 22, 33, 66.

3Keep only group sizes greater than 11

#6 Guess and Check 4.OA.B.4

The group must be bigger than 11, which rules out 1, 2, 3, 6, 11.

22, 33, 66

That leaves 3 group sizes.

4Confirm each surviving split works

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

Divide to see each one really does share out evenly.

66 \div 22 = 3,\quad 66 \div 33 = 2,\quad 66 \div 66 = 1

All 3 work, so there are 3 ways.

Answer: 3 ways

4 · Reviewdoes it hold up?

66 has 8 factors in all, and 5 of them are 11 or less, leaving 3.

Another way: Test each number from 12 up to 66 and keep the ones that divide 66; slower, but it gives the same 3.

Standardsmin grade 4

4.OA.B.4 Find all factor pairs and recognize multiples; determine prime or composite — Listing every factor of 66 in pairs and filtering by size.
3.OA.B.6 Understand division as an unknown-factor problem — Reading 'shares evenly with none left over' as exact division.

💡Takeaway. 'Shares evenly with none left over' is another way of saying 'is a factor' -- so list the factor pairs and nothing is missed.

Variant 9 hard answer: 3 ways

You want to share $72$ candies equally, with none left over, among more than $20$ students. In how many different ways can the candies be shared?

Show solution

1 · Understandwhat's really being asked

72 candies are split evenly among a group of more than 20 students, with nothing left over. We must count how many different group sizes make that possible.

Givens

There are 72 candies in total.
Everyone gets the same number, and none are left over.
The number of students is more than 20.

Unknowns

How many different group sizes work.

Constraints

Each student's share is a whole number of candies.

2 · Planchoose the strategy

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

'Equal shares with none left over' is the definition of a factor, so list the factors of 72 in pairs -- that way none is missed -- and then keep the ones big enough.

3 · Execute4 carry out the plan

1Turn 'equal shares, none left over' into divisors

#2 Make a Systematic List 3.OA.B.6

If 72 candies split evenly among some students, that number of students divides 72 exactly.

72 \div \square = \text{whole number}

The question is really about factors, not about candy.

2List all factor pairs of 72

#2 Make a Systematic List 4.OA.B.4

Pairing them up guarantees the list is complete: work upwards until the two halves of a pair meet.

72 = 1 \times 72, 2 \times 36, 3 \times 24, 4 \times 18, 6 \times 12, 8 \times 9

The factors are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72.

3Keep only group sizes greater than 20

#6 Guess and Check 4.OA.B.4

The group must be bigger than 20, which rules out 1, 2, 3, 4, 6, 8, 9, 12, 18.

24, 36, 72

That leaves 3 group sizes.

4Confirm each surviving split works

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

Divide to see each one really does share out evenly.

72 \div 24 = 3,\quad 72 \div 36 = 2,\quad 72 \div 72 = 1

All 3 work, so there are 3 ways.

Answer: 3 ways

4 · Reviewdoes it hold up?

72 has 12 factors in all, and 9 of them are 20 or less, leaving 3.

Another way: Test each number from 21 up to 72 and keep the ones that divide 72; slower, but it gives the same 3.

Standardsmin grade 4

4.OA.B.4 Find all factor pairs and recognize multiples; determine prime or composite — Listing every factor of 72 in pairs and filtering by size.
3.OA.B.6 Understand division as an unknown-factor problem — Reading 'shares evenly with none left over' as exact division.

💡Takeaway. 'Shares evenly with none left over' is another way of saying 'is a factor' -- so list the factor pairs and nothing is missed.

Variant 10 hard answer: 4 ways

You want to share $84$ candies equally, with none left over, among more than $18$ students. In how many different ways can the candies be shared?

Show solution

1 · Understandwhat's really being asked

84 candies are split evenly among a group of more than 18 students, with nothing left over. We must count how many different group sizes make that possible.

Givens

There are 84 candies in total.
Everyone gets the same number, and none are left over.
The number of students is more than 18.

Unknowns

How many different group sizes work.

Constraints

Each student's share is a whole number of candies.

2 · Planchoose the strategy

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

'Equal shares with none left over' is the definition of a factor, so list the factors of 84 in pairs -- that way none is missed -- and then keep the ones big enough.

3 · Execute4 carry out the plan

1Turn 'equal shares, none left over' into divisors

#2 Make a Systematic List 3.OA.B.6

If 84 candies split evenly among some students, that number of students divides 84 exactly.

84 \div \square = \text{whole number}

The question is really about factors, not about candy.

2List all factor pairs of 84

#2 Make a Systematic List 4.OA.B.4

Pairing them up guarantees the list is complete: work upwards until the two halves of a pair meet.

84 = 1 \times 84, 2 \times 42, 3 \times 28, 4 \times 21, 6 \times 14, 7 \times 12

The factors are 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84.

3Keep only group sizes greater than 18

#6 Guess and Check 4.OA.B.4

The group must be bigger than 18, which rules out 1, 2, 3, 4, 6, 7, 12, 14.

21, 28, 42, 84

That leaves 4 group sizes.

4Confirm each surviving split works

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

Divide to see each one really does share out evenly.

84 \div 21 = 4,\quad 84 \div 28 = 3,\quad 84 \div 42 = 2,\quad 84 \div 84 = 1

All 4 work, so there are 4 ways.

Answer: 4 ways

4 · Reviewdoes it hold up?

84 has 12 factors in all, and 8 of them are 18 or less, leaving 4.

Another way: Test each number from 19 up to 84 and keep the ones that divide 84; slower, but it gives the same 4.

Standardsmin grade 4

4.OA.B.4 Find all factor pairs and recognize multiples; determine prime or composite — Listing every factor of 84 in pairs and filtering by size.
3.OA.B.6 Understand division as an unknown-factor problem — Reading 'shares evenly with none left over' as exact division.

💡Takeaway. 'Shares evenly with none left over' is another way of saying 'is a factor' -- so list the factor pairs and nothing is missed.

Variant 11 hard answer: 3 ways

You want to share $90$ candies equally, with none left over, among more than $25$ students. In how many different ways can the candies be shared?

Show solution

1 · Understandwhat's really being asked

90 candies are split evenly among a group of more than 25 students, with nothing left over. We must count how many different group sizes make that possible.

Givens

There are 90 candies in total.
Everyone gets the same number, and none are left over.
The number of students is more than 25.

Unknowns

How many different group sizes work.

Constraints

Each student's share is a whole number of candies.

2 · Planchoose the strategy

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

'Equal shares with none left over' is the definition of a factor, so list the factors of 90 in pairs -- that way none is missed -- and then keep the ones big enough.

3 · Execute4 carry out the plan

1Turn 'equal shares, none left over' into divisors

#2 Make a Systematic List 3.OA.B.6

If 90 candies split evenly among some students, that number of students divides 90 exactly.

90 \div \square = \text{whole number}

The question is really about factors, not about candy.

2List all factor pairs of 90

#2 Make a Systematic List 4.OA.B.4

Pairing them up guarantees the list is complete: work upwards until the two halves of a pair meet.

90 = 1 \times 90, 2 \times 45, 3 \times 30, 5 \times 18, 6 \times 15, 9 \times 10

The factors are 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90.

3Keep only group sizes greater than 25

#6 Guess and Check 4.OA.B.4

The group must be bigger than 25, which rules out 1, 2, 3, 5, 6, 9, 10, 15, 18.

30, 45, 90

That leaves 3 group sizes.

4Confirm each surviving split works

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

Divide to see each one really does share out evenly.

90 \div 30 = 3,\quad 90 \div 45 = 2,\quad 90 \div 90 = 1

All 3 work, so there are 3 ways.

Answer: 3 ways

4 · Reviewdoes it hold up?

90 has 12 factors in all, and 9 of them are 25 or less, leaving 3.

Another way: Test each number from 26 up to 90 and keep the ones that divide 90; slower, but it gives the same 3.

Standardsmin grade 4

4.OA.B.4 Find all factor pairs and recognize multiples; determine prime or composite — Listing every factor of 90 in pairs and filtering by size.
3.OA.B.6 Understand division as an unknown-factor problem — Reading 'shares evenly with none left over' as exact division.

💡Takeaway. 'Shares evenly with none left over' is another way of saying 'is a factor' -- so list the factor pairs and nothing is missed.

Variant 12 hard answer: 3 ways

You want to share $96$ candies equally, with none left over, among more than $30$ students. In how many different ways can the candies be shared?

Show solution

1 · Understandwhat's really being asked

96 candies are split evenly among a group of more than 30 students, with nothing left over. We must count how many different group sizes make that possible.

Givens

There are 96 candies in total.
Everyone gets the same number, and none are left over.
The number of students is more than 30.

Unknowns

How many different group sizes work.

Constraints

Each student's share is a whole number of candies.

2 · Planchoose the strategy

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

'Equal shares with none left over' is the definition of a factor, so list the factors of 96 in pairs -- that way none is missed -- and then keep the ones big enough.

3 · Execute4 carry out the plan

1Turn 'equal shares, none left over' into divisors

#2 Make a Systematic List 3.OA.B.6

If 96 candies split evenly among some students, that number of students divides 96 exactly.

96 \div \square = \text{whole number}

The question is really about factors, not about candy.

2List all factor pairs of 96

#2 Make a Systematic List 4.OA.B.4

Pairing them up guarantees the list is complete: work upwards until the two halves of a pair meet.

96 = 1 \times 96, 2 \times 48, 3 \times 32, 4 \times 24, 6 \times 16, 8 \times 12

The factors are 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 96.

3Keep only group sizes greater than 30

#6 Guess and Check 4.OA.B.4

The group must be bigger than 30, which rules out 1, 2, 3, 4, 6, 8, 12, 16, 24.

32, 48, 96

That leaves 3 group sizes.

4Confirm each surviving split works

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

Divide to see each one really does share out evenly.

96 \div 32 = 3,\quad 96 \div 48 = 2,\quad 96 \div 96 = 1

All 3 work, so there are 3 ways.

Answer: 3 ways

4 · Reviewdoes it hold up?

96 has 12 factors in all, and 9 of them are 30 or less, leaving 3.

Another way: Test each number from 31 up to 96 and keep the ones that divide 96; slower, but it gives the same 3.

Standardsmin grade 4

4.OA.B.4 Find all factor pairs and recognize multiples; determine prime or composite — Listing every factor of 96 in pairs and filtering by size.
3.OA.B.6 Understand division as an unknown-factor problem — Reading 'shares evenly with none left over' as exact division.

💡Takeaway. 'Shares evenly with none left over' is another way of saying 'is a factor' -- so list the factor pairs and nothing is missed.