← Product of two numbers equals GCF times LCM · Divisibility and Remainder Reasoning

Product of two numbers equals GCF times LCM · 12 practice problems

6.NS.B.46.NS.B.2

Generated variants — 12

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

Variant 1 easy answer: 40

A certain number and $24$ have a greatest common factor of $8$ and a least common multiple of $120$ . Find the number.

Show solution

1 · Understandwhat's really being asked

One number is 24. Paired with a hidden number, their greatest common factor is 8 and their least common multiple is 120. We need the hidden number.

Givens

One of the two numbers is 24.
Their greatest common factor is 8.
Their least common multiple is 120.

Unknowns

The other number.

Constraints

Both numbers are multiples of 8, since 8 divides both.

2 · Planchoose the strategy

#9 Solve an Easier Related Problem · also uses: #5 Look for a Pattern#6 Guess and Check

Try a small pair where the gcf and lcm are easy to see, notice how they relate to the product, then use that relationship on the numbers given.

3 · Execute5 carry out the plan

1Discover the rule with a small pair

#9 Solve an Easier Related Problem 6.NS.B.4

Take 4 and 6: their greatest common factor is 2 and their least common multiple is 12. Compare 2 times 12 with 4 times 6.

2 \times 12 = 24,\quad 4 \times 6 = 24

The two products came out the same. That is not a coincidence.

2State the pattern

#5 Look for a Pattern 6.NS.B.4

For any two numbers, multiplying the greatest common factor by the least common multiple gives the same answer as multiplying the two numbers.

a \times b = \gcd(a, b) \times \operatorname{lcm}(a, b)

The gcf holds the shared factors and the lcm holds the rest, so together they account for every factor exactly once each.

3Plug in the known values

#9 Solve an Easier Related Problem 6.NS.B.4

The right-hand side is fully known: 8 times 120.

24 \times \square = 8 \times 120 = 960

The two numbers multiply to 960.

4Find the missing number

#9 Solve an Easier Related Problem 6.NS.B.2

Divide 960 by the number we know.

960 \div 24 = 40

The hidden number is 40.

5Check the candidate by guess and check

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

Verify directly that 40 and 24 really do have that greatest common factor and least common multiple.

\gcd(40, 24) = 8,\quad \operatorname{lcm}(40, 24) = 120

Both conditions hold, so the answer is confirmed.

Answer: 40

4 · Reviewdoes it hold up?

40 is a multiple of 8, as it must be, and 40 divides 120 -- both are necessary for the stated gcf and lcm.

Another way: Write both as 8 times something: 24 is 8 x 3, so the other is 8 x 5 where 3 and 5 share no factor. That gives 40 without the product rule.

Standardsmin grade 6

6.NS.B.4 Find greatest common factor and least common multiple of two numbers — Relating the greatest common factor and least common multiple to the product of the two numbers.
6.NS.B.2 Fluently divide multi-digit numbers using the standard algorithm — Dividing 960 by 24 to recover the missing number.

💡Takeaway. The greatest common factor and the least common multiple multiply to the same thing the two numbers do -- so any three of the four give you the fourth.

Variant 2 easy answer: 42

A certain number and $24$ have a greatest common factor of $6$ and a least common multiple of $168$ . Find the number.

Show solution

1 · Understandwhat's really being asked

One number is 24. Paired with a hidden number, their greatest common factor is 6 and their least common multiple is 168. We need the hidden number.

Givens

One of the two numbers is 24.
Their greatest common factor is 6.
Their least common multiple is 168.

Unknowns

The other number.

Constraints

Both numbers are multiples of 6, since 6 divides both.

2 · Planchoose the strategy

#9 Solve an Easier Related Problem · also uses: #5 Look for a Pattern#6 Guess and Check

Try a small pair where the gcf and lcm are easy to see, notice how they relate to the product, then use that relationship on the numbers given.

3 · Execute5 carry out the plan

1Discover the rule with a small pair

#9 Solve an Easier Related Problem 6.NS.B.4

Take 4 and 6: their greatest common factor is 2 and their least common multiple is 12. Compare 2 times 12 with 4 times 6.

2 \times 12 = 24,\quad 4 \times 6 = 24

The two products came out the same. That is not a coincidence.

2State the pattern

#5 Look for a Pattern 6.NS.B.4

For any two numbers, multiplying the greatest common factor by the least common multiple gives the same answer as multiplying the two numbers.

a \times b = \gcd(a, b) \times \operatorname{lcm}(a, b)

The gcf holds the shared factors and the lcm holds the rest, so together they account for every factor exactly once each.

3Plug in the known values

#9 Solve an Easier Related Problem 6.NS.B.4

The right-hand side is fully known: 6 times 168.

24 \times \square = 6 \times 168 = 1008

The two numbers multiply to 1008.

4Find the missing number

#9 Solve an Easier Related Problem 6.NS.B.2

Divide 1008 by the number we know.

1008 \div 24 = 42

The hidden number is 42.

5Check the candidate by guess and check

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

Verify directly that 42 and 24 really do have that greatest common factor and least common multiple.

\gcd(42, 24) = 6,\quad \operatorname{lcm}(42, 24) = 168

Both conditions hold, so the answer is confirmed.

Answer: 42

4 · Reviewdoes it hold up?

42 is a multiple of 6, as it must be, and 42 divides 168 -- both are necessary for the stated gcf and lcm.

Another way: Write both as 6 times something: 24 is 6 x 4, so the other is 6 x 7 where 4 and 7 share no factor. That gives 42 without the product rule.

Standardsmin grade 6

6.NS.B.4 Find greatest common factor and least common multiple of two numbers — Relating the greatest common factor and least common multiple to the product of the two numbers.
6.NS.B.2 Fluently divide multi-digit numbers using the standard algorithm — Dividing 1008 by 24 to recover the missing number.

💡Takeaway. The greatest common factor and the least common multiple multiply to the same thing the two numbers do -- so any three of the four give you the fourth.

Variant 3 easy answer: 60

A certain number and $36$ have a greatest common factor of $12$ and a least common multiple of $180$ . Find the number.

Show solution

1 · Understandwhat's really being asked

One number is 36. Paired with a hidden number, their greatest common factor is 12 and their least common multiple is 180. We need the hidden number.

Givens

One of the two numbers is 36.
Their greatest common factor is 12.
Their least common multiple is 180.

Unknowns

The other number.

Constraints

Both numbers are multiples of 12, since 12 divides both.

2 · Planchoose the strategy

#9 Solve an Easier Related Problem · also uses: #5 Look for a Pattern#6 Guess and Check

Try a small pair where the gcf and lcm are easy to see, notice how they relate to the product, then use that relationship on the numbers given.

3 · Execute5 carry out the plan

1Discover the rule with a small pair

#9 Solve an Easier Related Problem 6.NS.B.4

Take 4 and 6: their greatest common factor is 2 and their least common multiple is 12. Compare 2 times 12 with 4 times 6.

2 \times 12 = 24,\quad 4 \times 6 = 24

The two products came out the same. That is not a coincidence.

2State the pattern

#5 Look for a Pattern 6.NS.B.4

For any two numbers, multiplying the greatest common factor by the least common multiple gives the same answer as multiplying the two numbers.

a \times b = \gcd(a, b) \times \operatorname{lcm}(a, b)

The gcf holds the shared factors and the lcm holds the rest, so together they account for every factor exactly once each.

3Plug in the known values

#9 Solve an Easier Related Problem 6.NS.B.4

The right-hand side is fully known: 12 times 180.

36 \times \square = 12 \times 180 = 2160

The two numbers multiply to 2160.

4Find the missing number

#9 Solve an Easier Related Problem 6.NS.B.2

Divide 2160 by the number we know.

2160 \div 36 = 60

The hidden number is 60.

5Check the candidate by guess and check

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

Verify directly that 60 and 36 really do have that greatest common factor and least common multiple.

\gcd(60, 36) = 12,\quad \operatorname{lcm}(60, 36) = 180

Both conditions hold, so the answer is confirmed.

Answer: 60

4 · Reviewdoes it hold up?

60 is a multiple of 12, as it must be, and 60 divides 180 -- both are necessary for the stated gcf and lcm.

Another way: Write both as 12 times something: 36 is 12 x 3, so the other is 12 x 5 where 3 and 5 share no factor. That gives 60 without the product rule.

Standardsmin grade 6

6.NS.B.4 Find greatest common factor and least common multiple of two numbers — Relating the greatest common factor and least common multiple to the product of the two numbers.
6.NS.B.2 Fluently divide multi-digit numbers using the standard algorithm — Dividing 2160 by 36 to recover the missing number.

💡Takeaway. The greatest common factor and the least common multiple multiply to the same thing the two numbers do -- so any three of the four give you the fourth.

Variant 4 easy answer: 36

A certain number and $45$ have a greatest common factor of $9$ and a least common multiple of $180$ . Find the number.

Show solution

1 · Understandwhat's really being asked

One number is 45. Paired with a hidden number, their greatest common factor is 9 and their least common multiple is 180. We need the hidden number.

Givens

One of the two numbers is 45.
Their greatest common factor is 9.
Their least common multiple is 180.

Unknowns

The other number.

Constraints

Both numbers are multiples of 9, since 9 divides both.

2 · Planchoose the strategy

#9 Solve an Easier Related Problem · also uses: #5 Look for a Pattern#6 Guess and Check

Try a small pair where the gcf and lcm are easy to see, notice how they relate to the product, then use that relationship on the numbers given.

3 · Execute5 carry out the plan

1Discover the rule with a small pair

#9 Solve an Easier Related Problem 6.NS.B.4

Take 4 and 6: their greatest common factor is 2 and their least common multiple is 12. Compare 2 times 12 with 4 times 6.

2 \times 12 = 24,\quad 4 \times 6 = 24

The two products came out the same. That is not a coincidence.

2State the pattern

#5 Look for a Pattern 6.NS.B.4

For any two numbers, multiplying the greatest common factor by the least common multiple gives the same answer as multiplying the two numbers.

a \times b = \gcd(a, b) \times \operatorname{lcm}(a, b)

The gcf holds the shared factors and the lcm holds the rest, so together they account for every factor exactly once each.

3Plug in the known values

#9 Solve an Easier Related Problem 6.NS.B.4

The right-hand side is fully known: 9 times 180.

45 \times \square = 9 \times 180 = 1620

The two numbers multiply to 1620.

4Find the missing number

#9 Solve an Easier Related Problem 6.NS.B.2

Divide 1620 by the number we know.

1620 \div 45 = 36

The hidden number is 36.

5Check the candidate by guess and check

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

Verify directly that 36 and 45 really do have that greatest common factor and least common multiple.

\gcd(36, 45) = 9,\quad \operatorname{lcm}(36, 45) = 180

Both conditions hold, so the answer is confirmed.

Answer: 36

4 · Reviewdoes it hold up?

36 is a multiple of 9, as it must be, and 36 divides 180 -- both are necessary for the stated gcf and lcm.

Another way: Write both as 9 times something: 45 is 9 x 5, so the other is 9 x 4 where 5 and 4 share no factor. That gives 36 without the product rule.

Standardsmin grade 6

6.NS.B.4 Find greatest common factor and least common multiple of two numbers — Relating the greatest common factor and least common multiple to the product of the two numbers.
6.NS.B.2 Fluently divide multi-digit numbers using the standard algorithm — Dividing 1620 by 45 to recover the missing number.

💡Takeaway. The greatest common factor and the least common multiple multiply to the same thing the two numbers do -- so any three of the four give you the fourth.

Variant 5 medium answer: 90

A certain number and $36$ have a greatest common factor of $18$ and a least common multiple of $180$ . Find the number.

Show solution

1 · Understandwhat's really being asked

One number is 36. Paired with a hidden number, their greatest common factor is 18 and their least common multiple is 180. We need the hidden number.

Givens

One of the two numbers is 36.
Their greatest common factor is 18.
Their least common multiple is 180.

Unknowns

The other number.

Constraints

Both numbers are multiples of 18, since 18 divides both.

2 · Planchoose the strategy

#9 Solve an Easier Related Problem · also uses: #5 Look for a Pattern#6 Guess and Check

Try a small pair where the gcf and lcm are easy to see, notice how they relate to the product, then use that relationship on the numbers given.

3 · Execute5 carry out the plan

1Discover the rule with a small pair

#9 Solve an Easier Related Problem 6.NS.B.4

Take 4 and 6: their greatest common factor is 2 and their least common multiple is 12. Compare 2 times 12 with 4 times 6.

2 \times 12 = 24,\quad 4 \times 6 = 24

The two products came out the same. That is not a coincidence.

2State the pattern

#5 Look for a Pattern 6.NS.B.4

For any two numbers, multiplying the greatest common factor by the least common multiple gives the same answer as multiplying the two numbers.

a \times b = \gcd(a, b) \times \operatorname{lcm}(a, b)

The gcf holds the shared factors and the lcm holds the rest, so together they account for every factor exactly once each.

3Plug in the known values

#9 Solve an Easier Related Problem 6.NS.B.4

The right-hand side is fully known: 18 times 180.

36 \times \square = 18 \times 180 = 3240

The two numbers multiply to 3240.

4Find the missing number

#9 Solve an Easier Related Problem 6.NS.B.2

Divide 3240 by the number we know.

3240 \div 36 = 90

The hidden number is 90.

5Check the candidate by guess and check

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

Verify directly that 90 and 36 really do have that greatest common factor and least common multiple.

\gcd(90, 36) = 18,\quad \operatorname{lcm}(90, 36) = 180

Both conditions hold, so the answer is confirmed.

Answer: 90

4 · Reviewdoes it hold up?

90 is a multiple of 18, as it must be, and 90 divides 180 -- both are necessary for the stated gcf and lcm.

Another way: Write both as 18 times something: 36 is 18 x 2, so the other is 18 x 5 where 2 and 5 share no factor. That gives 90 without the product rule.

Standardsmin grade 6

6.NS.B.4 Find greatest common factor and least common multiple of two numbers — Relating the greatest common factor and least common multiple to the product of the two numbers.
6.NS.B.2 Fluently divide multi-digit numbers using the standard algorithm — Dividing 3240 by 36 to recover the missing number.

💡Takeaway. The greatest common factor and the least common multiple multiply to the same thing the two numbers do -- so any three of the four give you the fourth.

Variant 6 medium answer: 105

A certain number and $30$ have a greatest common factor of $15$ and a least common multiple of $210$ . Find the number.

Show solution

1 · Understandwhat's really being asked

One number is 30. Paired with a hidden number, their greatest common factor is 15 and their least common multiple is 210. We need the hidden number.

Givens

One of the two numbers is 30.
Their greatest common factor is 15.
Their least common multiple is 210.

Unknowns

The other number.

Constraints

Both numbers are multiples of 15, since 15 divides both.

2 · Planchoose the strategy

#9 Solve an Easier Related Problem · also uses: #5 Look for a Pattern#6 Guess and Check

Try a small pair where the gcf and lcm are easy to see, notice how they relate to the product, then use that relationship on the numbers given.

3 · Execute5 carry out the plan

1Discover the rule with a small pair

#9 Solve an Easier Related Problem 6.NS.B.4

Take 4 and 6: their greatest common factor is 2 and their least common multiple is 12. Compare 2 times 12 with 4 times 6.

2 \times 12 = 24,\quad 4 \times 6 = 24

The two products came out the same. That is not a coincidence.

2State the pattern

#5 Look for a Pattern 6.NS.B.4

For any two numbers, multiplying the greatest common factor by the least common multiple gives the same answer as multiplying the two numbers.

a \times b = \gcd(a, b) \times \operatorname{lcm}(a, b)

The gcf holds the shared factors and the lcm holds the rest, so together they account for every factor exactly once each.

3Plug in the known values

#9 Solve an Easier Related Problem 6.NS.B.4

The right-hand side is fully known: 15 times 210.

30 \times \square = 15 \times 210 = 3150

The two numbers multiply to 3150.

4Find the missing number

#9 Solve an Easier Related Problem 6.NS.B.2

Divide 3150 by the number we know.

3150 \div 30 = 105

The hidden number is 105.

5Check the candidate by guess and check

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

Verify directly that 105 and 30 really do have that greatest common factor and least common multiple.

\gcd(105, 30) = 15,\quad \operatorname{lcm}(105, 30) = 210

Both conditions hold, so the answer is confirmed.

Answer: 105

4 · Reviewdoes it hold up?

105 is a multiple of 15, as it must be, and 105 divides 210 -- both are necessary for the stated gcf and lcm.

Another way: Write both as 15 times something: 30 is 15 x 2, so the other is 15 x 7 where 2 and 7 share no factor. That gives 105 without the product rule.

Standardsmin grade 6

6.NS.B.4 Find greatest common factor and least common multiple of two numbers — Relating the greatest common factor and least common multiple to the product of the two numbers.
6.NS.B.2 Fluently divide multi-digit numbers using the standard algorithm — Dividing 3150 by 30 to recover the missing number.

💡Takeaway. The greatest common factor and the least common multiple multiply to the same thing the two numbers do -- so any three of the four give you the fourth.

Variant 7 medium answer: 70

A certain number and $30$ have a greatest common factor of $10$ and a least common multiple of $210$ . Find the number.

Show solution

1 · Understandwhat's really being asked

One number is 30. Paired with a hidden number, their greatest common factor is 10 and their least common multiple is 210. We need the hidden number.

Givens

One of the two numbers is 30.
Their greatest common factor is 10.
Their least common multiple is 210.

Unknowns

The other number.

Constraints

Both numbers are multiples of 10, since 10 divides both.

2 · Planchoose the strategy

#9 Solve an Easier Related Problem · also uses: #5 Look for a Pattern#6 Guess and Check

Try a small pair where the gcf and lcm are easy to see, notice how they relate to the product, then use that relationship on the numbers given.

3 · Execute5 carry out the plan

1Discover the rule with a small pair

#9 Solve an Easier Related Problem 6.NS.B.4

Take 4 and 6: their greatest common factor is 2 and their least common multiple is 12. Compare 2 times 12 with 4 times 6.

2 \times 12 = 24,\quad 4 \times 6 = 24

The two products came out the same. That is not a coincidence.

2State the pattern

#5 Look for a Pattern 6.NS.B.4

For any two numbers, multiplying the greatest common factor by the least common multiple gives the same answer as multiplying the two numbers.

a \times b = \gcd(a, b) \times \operatorname{lcm}(a, b)

The gcf holds the shared factors and the lcm holds the rest, so together they account for every factor exactly once each.

3Plug in the known values

#9 Solve an Easier Related Problem 6.NS.B.4

The right-hand side is fully known: 10 times 210.

30 \times \square = 10 \times 210 = 2100

The two numbers multiply to 2100.

4Find the missing number

#9 Solve an Easier Related Problem 6.NS.B.2

Divide 2100 by the number we know.

2100 \div 30 = 70

The hidden number is 70.

5Check the candidate by guess and check

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

Verify directly that 70 and 30 really do have that greatest common factor and least common multiple.

\gcd(70, 30) = 10,\quad \operatorname{lcm}(70, 30) = 210

Both conditions hold, so the answer is confirmed.

Answer: 70

4 · Reviewdoes it hold up?

70 is a multiple of 10, as it must be, and 70 divides 210 -- both are necessary for the stated gcf and lcm.

Another way: Write both as 10 times something: 30 is 10 x 3, so the other is 10 x 7 where 3 and 7 share no factor. That gives 70 without the product rule.

Standardsmin grade 6

6.NS.B.4 Find greatest common factor and least common multiple of two numbers — Relating the greatest common factor and least common multiple to the product of the two numbers.
6.NS.B.2 Fluently divide multi-digit numbers using the standard algorithm — Dividing 2100 by 30 to recover the missing number.

💡Takeaway. The greatest common factor and the least common multiple multiply to the same thing the two numbers do -- so any three of the four give you the fourth.

Variant 8 medium answer: 70

A certain number and $42$ have a greatest common factor of $14$ and a least common multiple of $210$ . Find the number.

Show solution

1 · Understandwhat's really being asked

One number is 42. Paired with a hidden number, their greatest common factor is 14 and their least common multiple is 210. We need the hidden number.

Givens

One of the two numbers is 42.
Their greatest common factor is 14.
Their least common multiple is 210.

Unknowns

The other number.

Constraints

Both numbers are multiples of 14, since 14 divides both.

2 · Planchoose the strategy

#9 Solve an Easier Related Problem · also uses: #5 Look for a Pattern#6 Guess and Check

Try a small pair where the gcf and lcm are easy to see, notice how they relate to the product, then use that relationship on the numbers given.

3 · Execute5 carry out the plan

1Discover the rule with a small pair

#9 Solve an Easier Related Problem 6.NS.B.4

Take 4 and 6: their greatest common factor is 2 and their least common multiple is 12. Compare 2 times 12 with 4 times 6.

2 \times 12 = 24,\quad 4 \times 6 = 24

The two products came out the same. That is not a coincidence.

2State the pattern

#5 Look for a Pattern 6.NS.B.4

For any two numbers, multiplying the greatest common factor by the least common multiple gives the same answer as multiplying the two numbers.

a \times b = \gcd(a, b) \times \operatorname{lcm}(a, b)

The gcf holds the shared factors and the lcm holds the rest, so together they account for every factor exactly once each.

3Plug in the known values

#9 Solve an Easier Related Problem 6.NS.B.4

The right-hand side is fully known: 14 times 210.

42 \times \square = 14 \times 210 = 2940

The two numbers multiply to 2940.

4Find the missing number

#9 Solve an Easier Related Problem 6.NS.B.2

Divide 2940 by the number we know.

2940 \div 42 = 70

The hidden number is 70.

5Check the candidate by guess and check

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

Verify directly that 70 and 42 really do have that greatest common factor and least common multiple.

\gcd(70, 42) = 14,\quad \operatorname{lcm}(70, 42) = 210

Both conditions hold, so the answer is confirmed.

Answer: 70

4 · Reviewdoes it hold up?

70 is a multiple of 14, as it must be, and 70 divides 210 -- both are necessary for the stated gcf and lcm.

Another way: Write both as 14 times something: 42 is 14 x 3, so the other is 14 x 5 where 3 and 5 share no factor. That gives 70 without the product rule.

Standardsmin grade 6

6.NS.B.4 Find greatest common factor and least common multiple of two numbers — Relating the greatest common factor and least common multiple to the product of the two numbers.
6.NS.B.2 Fluently divide multi-digit numbers using the standard algorithm — Dividing 2940 by 42 to recover the missing number.

💡Takeaway. The greatest common factor and the least common multiple multiply to the same thing the two numbers do -- so any three of the four give you the fourth.

Variant 9 hard answer: 63

A certain number and $28$ have a greatest common factor of $7$ and a least common multiple of $252$ . Find the number.

Show solution

1 · Understandwhat's really being asked

One number is 28. Paired with a hidden number, their greatest common factor is 7 and their least common multiple is 252. We need the hidden number.

Givens

One of the two numbers is 28.
Their greatest common factor is 7.
Their least common multiple is 252.

Unknowns

The other number.

Constraints

Both numbers are multiples of 7, since 7 divides both.

2 · Planchoose the strategy

#9 Solve an Easier Related Problem · also uses: #5 Look for a Pattern#6 Guess and Check

Try a small pair where the gcf and lcm are easy to see, notice how they relate to the product, then use that relationship on the numbers given.

3 · Execute5 carry out the plan

1Discover the rule with a small pair

#9 Solve an Easier Related Problem 6.NS.B.4

Take 4 and 6: their greatest common factor is 2 and their least common multiple is 12. Compare 2 times 12 with 4 times 6.

2 \times 12 = 24,\quad 4 \times 6 = 24

The two products came out the same. That is not a coincidence.

2State the pattern

#5 Look for a Pattern 6.NS.B.4

For any two numbers, multiplying the greatest common factor by the least common multiple gives the same answer as multiplying the two numbers.

a \times b = \gcd(a, b) \times \operatorname{lcm}(a, b)

The gcf holds the shared factors and the lcm holds the rest, so together they account for every factor exactly once each.

3Plug in the known values

#9 Solve an Easier Related Problem 6.NS.B.4

The right-hand side is fully known: 7 times 252.

28 \times \square = 7 \times 252 = 1764

The two numbers multiply to 1764.

4Find the missing number

#9 Solve an Easier Related Problem 6.NS.B.2

Divide 1764 by the number we know.

1764 \div 28 = 63

The hidden number is 63.

5Check the candidate by guess and check

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

Verify directly that 63 and 28 really do have that greatest common factor and least common multiple.

\gcd(63, 28) = 7,\quad \operatorname{lcm}(63, 28) = 252

Both conditions hold, so the answer is confirmed.

Answer: 63

4 · Reviewdoes it hold up?

63 is a multiple of 7, as it must be, and 63 divides 252 -- both are necessary for the stated gcf and lcm.

Another way: Write both as 7 times something: 28 is 7 x 4, so the other is 7 x 9 where 4 and 9 share no factor. That gives 63 without the product rule.

Standardsmin grade 6

6.NS.B.4 Find greatest common factor and least common multiple of two numbers — Relating the greatest common factor and least common multiple to the product of the two numbers.
6.NS.B.2 Fluently divide multi-digit numbers using the standard algorithm — Dividing 1764 by 28 to recover the missing number.

💡Takeaway. The greatest common factor and the least common multiple multiply to the same thing the two numbers do -- so any three of the four give you the fourth.

Variant 10 hard answer: 88

A certain number and $33$ have a greatest common factor of $11$ and a least common multiple of $264$ . Find the number.

Show solution

1 · Understandwhat's really being asked

One number is 33. Paired with a hidden number, their greatest common factor is 11 and their least common multiple is 264. We need the hidden number.

Givens

One of the two numbers is 33.
Their greatest common factor is 11.
Their least common multiple is 264.

Unknowns

The other number.

Constraints

Both numbers are multiples of 11, since 11 divides both.

2 · Planchoose the strategy

#9 Solve an Easier Related Problem · also uses: #5 Look for a Pattern#6 Guess and Check

Try a small pair where the gcf and lcm are easy to see, notice how they relate to the product, then use that relationship on the numbers given.

3 · Execute5 carry out the plan

1Discover the rule with a small pair

#9 Solve an Easier Related Problem 6.NS.B.4

Take 4 and 6: their greatest common factor is 2 and their least common multiple is 12. Compare 2 times 12 with 4 times 6.

2 \times 12 = 24,\quad 4 \times 6 = 24

The two products came out the same. That is not a coincidence.

2State the pattern

#5 Look for a Pattern 6.NS.B.4

For any two numbers, multiplying the greatest common factor by the least common multiple gives the same answer as multiplying the two numbers.

a \times b = \gcd(a, b) \times \operatorname{lcm}(a, b)

The gcf holds the shared factors and the lcm holds the rest, so together they account for every factor exactly once each.

3Plug in the known values

#9 Solve an Easier Related Problem 6.NS.B.4

The right-hand side is fully known: 11 times 264.

33 \times \square = 11 \times 264 = 2904

The two numbers multiply to 2904.

4Find the missing number

#9 Solve an Easier Related Problem 6.NS.B.2

Divide 2904 by the number we know.

2904 \div 33 = 88

The hidden number is 88.

5Check the candidate by guess and check

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

Verify directly that 88 and 33 really do have that greatest common factor and least common multiple.

\gcd(88, 33) = 11,\quad \operatorname{lcm}(88, 33) = 264

Both conditions hold, so the answer is confirmed.

Answer: 88

4 · Reviewdoes it hold up?

88 is a multiple of 11, as it must be, and 88 divides 264 -- both are necessary for the stated gcf and lcm.

Another way: Write both as 11 times something: 33 is 11 x 3, so the other is 11 x 8 where 3 and 8 share no factor. That gives 88 without the product rule.

Standardsmin grade 6

6.NS.B.4 Find greatest common factor and least common multiple of two numbers — Relating the greatest common factor and least common multiple to the product of the two numbers.
6.NS.B.2 Fluently divide multi-digit numbers using the standard algorithm — Dividing 2904 by 33 to recover the missing number.

💡Takeaway. The greatest common factor and the least common multiple multiply to the same thing the two numbers do -- so any three of the four give you the fourth.

Variant 11 hard answer: 72

A certain number and $120$ have a greatest common factor of $24$ and a least common multiple of $360$ . Find the number.

Show solution

1 · Understandwhat's really being asked

One number is 120. Paired with a hidden number, their greatest common factor is 24 and their least common multiple is 360. We need the hidden number.

Givens

One of the two numbers is 120.
Their greatest common factor is 24.
Their least common multiple is 360.

Unknowns

The other number.

Constraints

Both numbers are multiples of 24, since 24 divides both.

2 · Planchoose the strategy

#9 Solve an Easier Related Problem · also uses: #5 Look for a Pattern#6 Guess and Check

Try a small pair where the gcf and lcm are easy to see, notice how they relate to the product, then use that relationship on the numbers given.

3 · Execute5 carry out the plan

1Discover the rule with a small pair

#9 Solve an Easier Related Problem 6.NS.B.4

Take 4 and 6: their greatest common factor is 2 and their least common multiple is 12. Compare 2 times 12 with 4 times 6.

2 \times 12 = 24,\quad 4 \times 6 = 24

The two products came out the same. That is not a coincidence.

2State the pattern

#5 Look for a Pattern 6.NS.B.4

For any two numbers, multiplying the greatest common factor by the least common multiple gives the same answer as multiplying the two numbers.

a \times b = \gcd(a, b) \times \operatorname{lcm}(a, b)

The gcf holds the shared factors and the lcm holds the rest, so together they account for every factor exactly once each.

3Plug in the known values

#9 Solve an Easier Related Problem 6.NS.B.4

The right-hand side is fully known: 24 times 360.

120 \times \square = 24 \times 360 = 8640

The two numbers multiply to 8640.

4Find the missing number

#9 Solve an Easier Related Problem 6.NS.B.2

Divide 8640 by the number we know.

8640 \div 120 = 72

The hidden number is 72.

5Check the candidate by guess and check

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

Verify directly that 72 and 120 really do have that greatest common factor and least common multiple.

\gcd(72, 120) = 24,\quad \operatorname{lcm}(72, 120) = 360

Both conditions hold, so the answer is confirmed.

Answer: 72

4 · Reviewdoes it hold up?

72 is a multiple of 24, as it must be, and 72 divides 360 -- both are necessary for the stated gcf and lcm.

Another way: Write both as 24 times something: 120 is 24 x 5, so the other is 24 x 3 where 5 and 3 share no factor. That gives 72 without the product rule.

Standardsmin grade 6

6.NS.B.4 Find greatest common factor and least common multiple of two numbers — Relating the greatest common factor and least common multiple to the product of the two numbers.
6.NS.B.2 Fluently divide multi-digit numbers using the standard algorithm — Dividing 8640 by 120 to recover the missing number.

💡Takeaway. The greatest common factor and the least common multiple multiply to the same thing the two numbers do -- so any three of the four give you the fourth.

Variant 12 hard answer: 140

A certain number and $60$ have a greatest common factor of $20$ and a least common multiple of $420$ . Find the number.

Show solution

1 · Understandwhat's really being asked

One number is 60. Paired with a hidden number, their greatest common factor is 20 and their least common multiple is 420. We need the hidden number.

Givens

One of the two numbers is 60.
Their greatest common factor is 20.
Their least common multiple is 420.

Unknowns

The other number.

Constraints

Both numbers are multiples of 20, since 20 divides both.

2 · Planchoose the strategy

#9 Solve an Easier Related Problem · also uses: #5 Look for a Pattern#6 Guess and Check

Try a small pair where the gcf and lcm are easy to see, notice how they relate to the product, then use that relationship on the numbers given.

3 · Execute5 carry out the plan

1Discover the rule with a small pair

#9 Solve an Easier Related Problem 6.NS.B.4

Take 4 and 6: their greatest common factor is 2 and their least common multiple is 12. Compare 2 times 12 with 4 times 6.

2 \times 12 = 24,\quad 4 \times 6 = 24

The two products came out the same. That is not a coincidence.

2State the pattern

#5 Look for a Pattern 6.NS.B.4

For any two numbers, multiplying the greatest common factor by the least common multiple gives the same answer as multiplying the two numbers.

a \times b = \gcd(a, b) \times \operatorname{lcm}(a, b)

The gcf holds the shared factors and the lcm holds the rest, so together they account for every factor exactly once each.

3Plug in the known values

#9 Solve an Easier Related Problem 6.NS.B.4

The right-hand side is fully known: 20 times 420.

60 \times \square = 20 \times 420 = 8400

The two numbers multiply to 8400.

4Find the missing number

#9 Solve an Easier Related Problem 6.NS.B.2

Divide 8400 by the number we know.

8400 \div 60 = 140

The hidden number is 140.

5Check the candidate by guess and check

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

Verify directly that 140 and 60 really do have that greatest common factor and least common multiple.

\gcd(140, 60) = 20,\quad \operatorname{lcm}(140, 60) = 420

Both conditions hold, so the answer is confirmed.

Answer: 140

4 · Reviewdoes it hold up?

140 is a multiple of 20, as it must be, and 140 divides 420 -- both are necessary for the stated gcf and lcm.

Another way: Write both as 20 times something: 60 is 20 x 3, so the other is 20 x 7 where 3 and 7 share no factor. That gives 140 without the product rule.

Standardsmin grade 6

6.NS.B.4 Find greatest common factor and least common multiple of two numbers — Relating the greatest common factor and least common multiple to the product of the two numbers.
6.NS.B.2 Fluently divide multi-digit numbers using the standard algorithm — Dividing 8400 by 60 to recover the missing number.

💡Takeaway. The greatest common factor and the least common multiple multiply to the same thing the two numbers do -- so any three of the four give you the fourth.