Practice · Let the LCM shrink the candidate solutions · Sensim Math

Variant 1 easy

The natural numbers $x$ and $y$ satisfy the linear equation $3x-y=10$ , and the least common multiple of $x$ and $y$ is $5$ . Find the value of $x+y$ .

Your answer

Show solution

1 · Understandwhat's really being asked

The equation $3x-y=10$ has infinitely many natural-number solutions. Among them we want the one where $x$ and $y$ have least common multiple $5$ , and then we report $x+y$ .

Givens

$x$ and $y$ are natural numbers.
$3x-y=10$ .
The least common multiple of $x$ and $y$ is $5$ .

Unknowns

The value of $x+y$ .

Constraints

Both $x$ and $y$ must be natural numbers (positive integers).

2 · Planchoose the strategy

#3 Eliminate Possibilities · also uses: #2 Make a Systematic List#13 Convert to Algebra

Listing solutions of $3x-y=10$ never ends, so start from the other condition instead. A number always divides any common multiple of itself, so $x$ and $y$ must both divide $5$ . That leaves only four pairs to test against the equation.

3 · Execute3 carry out the plan

1Turn the LCM condition into a short list

#3 Eliminate Possibilities 6.NS.B.4

Each of

x

and

y

divides their least common multiple, which is

5

. The only natural numbers that divide

5

are

1

and

5

.

x \in \{1, 5\}, \quad y \in \{1, 5\}

A prime LCM leaves very few choices for each number.

2Test the four pairs in the equation

#2 Make a Systematic List 6.EE.B.5

Check

3x-y

for every pair. Only

(5, 5)

gives

10

, and its least common multiple really is

5

.

3 \times 5 - 1 \times 5 = 15 - 5 = 10

The other pairs give

3-1=2

,

3-5=-2

and

15-1=14

, so they fail.

3Add the two values

#13 Convert to Algebra 6.EE.B.5

With

x=5

and

y=5

, the requested sum follows.

x + y = 5 + 5 = 10

The answer is a single number, as expected.

Answer:

10

4 · Reviewdoes it hold up?

Substitute back: $3 \times 5 - 1 \times 5 = 10$ , and the least common multiple of $5$ and $5$ is $5$ . Both conditions hold, so $x+y=10$ .

Another way: List the natural solutions of $3x-y=10$ directly: each next one has $x$ larger by $1$ , giving $(4, 2)$ , $(5, 5)$ , $(6, 8)$ , $(7, 11)$ , $\ldots$ Their least common multiples are $4$ , $5$ , $24$ , $77$ , $\ldots$ and only $(5, 5)$ has least common multiple $5$ .

Standardsmin grade 6

6.NS.B.4 Find GCF and LCM; distributive property with a common factor — Using that each number divides its least common multiple, so $x$ and $y$ can only be $1$ or $5$.
6.EE.B.5 Solve equations/inequalities by finding values that make them true — Substituting each candidate pair to see which one makes $3x-y=10$ true.

💡Takeaway. When one condition has endless answers, start from the other condition that allows only a few.

Variant 2 easy

The natural numbers $x$ and $y$ satisfy the linear equation $3x-2y=7$ , and the least common multiple of $x$ and $y$ is $7$ . Find the value of $x+y$ .

Your answer

Show solution

1 · Understandwhat's really being asked

The equation $3x-2y=7$ has infinitely many natural-number solutions. Among them we want the one where $x$ and $y$ have least common multiple $7$ , and then we report $x+y$ .

Givens

$x$ and $y$ are natural numbers.
$3x-2y=7$ .
The least common multiple of $x$ and $y$ is $7$ .

Unknowns

The value of $x+y$ .

Constraints

Both $x$ and $y$ must be natural numbers (positive integers).

2 · Planchoose the strategy

#3 Eliminate Possibilities · also uses: #2 Make a Systematic List#13 Convert to Algebra

Listing solutions of $3x-2y=7$ never ends, so start from the other condition instead. A number always divides any common multiple of itself, so $x$ and $y$ must both divide $7$ . That leaves only four pairs to test against the equation.

3 · Execute3 carry out the plan

1Turn the LCM condition into a short list

#3 Eliminate Possibilities 6.NS.B.4

Each of

x

and

y

divides their least common multiple, which is

7

. The only natural numbers that divide

7

are

1

and

7

.

x \in \{1, 7\}, \quad y \in \{1, 7\}

A prime LCM leaves very few choices for each number.

2Test the four pairs in the equation

#2 Make a Systematic List 6.EE.B.5

Check

3x-2y

for every pair. Only

(7, 7)

gives

7

, and its least common multiple really is

7

.

3 \times 7 - 2 \times 7 = 21 - 14 = 7

The other pairs give

3-2=1

,

3-14=-11

and

21-2=19

, so they fail.

3Add the two values

#13 Convert to Algebra 6.EE.B.5

With

x=7

and

y=7

, the requested sum follows.

x + y = 7 + 7 = 14

The answer is a single number, as expected.

Answer:

14

4 · Reviewdoes it hold up?

Substitute back: $3 \times 7 - 2 \times 7 = 7$ , and the least common multiple of $7$ and $7$ is $7$ . Both conditions hold, so $x+y=14$ .

Another way: List the natural solutions of $3x-2y=7$ directly: each next one has $x$ larger by $2$ , giving $(3, 1)$ , $(5, 4)$ , $(7, 7)$ , $(9, 10)$ , $\ldots$ Their least common multiples are $3$ , $20$ , $7$ , $90$ , $\ldots$ and only $(7, 7)$ has least common multiple $7$ .

Standardsmin grade 6

6.NS.B.4 Find GCF and LCM; distributive property with a common factor — Using that each number divides its least common multiple, so $x$ and $y$ can only be $1$ or $7$.
6.EE.B.5 Solve equations/inequalities by finding values that make them true — Substituting each candidate pair to see which one makes $3x-2y=7$ true.

💡Takeaway. When one condition has endless answers, start from the other condition that allows only a few.

Variant 3 easy

The natural numbers $x$ and $y$ satisfy the linear equation $4x-3y=11$ , and the least common multiple of $x$ and $y$ is $11$ . Find the value of $x+y$ .

Your answer

Show solution

1 · Understandwhat's really being asked

The equation $4x-3y=11$ has infinitely many natural-number solutions. Among them we want the one where $x$ and $y$ have least common multiple $11$ , and then we report $x+y$ .

Givens

$x$ and $y$ are natural numbers.
$4x-3y=11$ .
The least common multiple of $x$ and $y$ is $11$ .

Unknowns

The value of $x+y$ .

Constraints

Both $x$ and $y$ must be natural numbers (positive integers).

2 · Planchoose the strategy

#3 Eliminate Possibilities · also uses: #2 Make a Systematic List#13 Convert to Algebra

Listing solutions of $4x-3y=11$ never ends, so start from the other condition instead. A number always divides any common multiple of itself, so $x$ and $y$ must both divide $11$ . That leaves only four pairs to test against the equation.

3 · Execute3 carry out the plan

1Turn the LCM condition into a short list

#3 Eliminate Possibilities 6.NS.B.4

Each of

x

and

y

divides their least common multiple, which is

11

. The only natural numbers that divide

11

are

1

and

11

.

x \in \{1, 11\}, \quad y \in \{1, 11\}

A prime LCM leaves very few choices for each number.

2Test the four pairs in the equation

#2 Make a Systematic List 6.EE.B.5

Check

4x-3y

for every pair. Only

(11, 11)

gives

11

, and its least common multiple really is

11

.

4 \times 11 - 3 \times 11 = 44 - 33 = 11

The other pairs give

4-3=1

,

4-33=-29

and

44-3=41

, so they fail.

3Add the two values

#13 Convert to Algebra 6.EE.B.5

With

x=11

and

y=11

, the requested sum follows.

x + y = 11 + 11 = 22

The answer is a single number, as expected.

Answer:

22

4 · Reviewdoes it hold up?

Substitute back: $4 \times 11 - 3 \times 11 = 11$ , and the least common multiple of $11$ and $11$ is $11$ . Both conditions hold, so $x+y=22$ .

Another way: List the natural solutions of $4x-3y=11$ directly: each next one has $x$ larger by $3$ , giving $(5, 3)$ , $(8, 7)$ , $(11, 11)$ , $(14, 15)$ , $\ldots$ Their least common multiples are $15$ , $56$ , $11$ , $210$ , $\ldots$ and only $(11, 11)$ has least common multiple $11$ .

Standardsmin grade 6

6.NS.B.4 Find GCF and LCM; distributive property with a common factor — Using that each number divides its least common multiple, so $x$ and $y$ can only be $1$ or $11$.
6.EE.B.5 Solve equations/inequalities by finding values that make them true — Substituting each candidate pair to see which one makes $4x-3y=11$ true.

💡Takeaway. When one condition has endless answers, start from the other condition that allows only a few.

Variant 4 easy

The natural numbers $x$ and $y$ satisfy the linear equation $4x-y=21$ , and the least common multiple of $x$ and $y$ is $7$ . Find the value of $x+y$ .

Your answer

Show solution

1 · Understandwhat's really being asked

The equation $4x-y=21$ has infinitely many natural-number solutions. Among them we want the one where $x$ and $y$ have least common multiple $7$ , and then we report $x+y$ .

Givens

$x$ and $y$ are natural numbers.
$4x-y=21$ .
The least common multiple of $x$ and $y$ is $7$ .

Unknowns

The value of $x+y$ .

Constraints

Both $x$ and $y$ must be natural numbers (positive integers).

2 · Planchoose the strategy

#3 Eliminate Possibilities · also uses: #2 Make a Systematic List#13 Convert to Algebra

Listing solutions of $4x-y=21$ never ends, so start from the other condition instead. A number always divides any common multiple of itself, so $x$ and $y$ must both divide $7$ . That leaves only four pairs to test against the equation.

3 · Execute3 carry out the plan

1Turn the LCM condition into a short list

#3 Eliminate Possibilities 6.NS.B.4

Each of

x

and

y

divides their least common multiple, which is

7

. The only natural numbers that divide

7

are

1

and

7

.

x \in \{1, 7\}, \quad y \in \{1, 7\}

A prime LCM leaves very few choices for each number.

2Test the four pairs in the equation

#2 Make a Systematic List 6.EE.B.5

Check

4x-y

for every pair. Only

(7, 7)

gives

21

, and its least common multiple really is

7

.

4 \times 7 - 1 \times 7 = 28 - 7 = 21

The other pairs give

4-1=3

,

4-7=-3

and

28-1=27

, so they fail.

3Add the two values

#13 Convert to Algebra 6.EE.B.5

With

x=7

and

y=7

, the requested sum follows.

x + y = 7 + 7 = 14

The answer is a single number, as expected.

Answer:

14

4 · Reviewdoes it hold up?

Substitute back: $4 \times 7 - 1 \times 7 = 21$ , and the least common multiple of $7$ and $7$ is $7$ . Both conditions hold, so $x+y=14$ .

Another way: List the natural solutions of $4x-y=21$ directly: each next one has $x$ larger by $1$ , giving $(6, 3)$ , $(7, 7)$ , $(8, 11)$ , $(9, 15)$ , $\ldots$ Their least common multiples are $6$ , $7$ , $88$ , $45$ , $\ldots$ and only $(7, 7)$ has least common multiple $7$ .

Standardsmin grade 6

6.NS.B.4 Find GCF and LCM; distributive property with a common factor — Using that each number divides its least common multiple, so $x$ and $y$ can only be $1$ or $7$.
6.EE.B.5 Solve equations/inequalities by finding values that make them true — Substituting each candidate pair to see which one makes $4x-y=21$ true.

💡Takeaway. When one condition has endless answers, start from the other condition that allows only a few.

Variant 5 medium

The natural numbers $x$ and $y$ satisfy the linear equation $6x-5y=17$ , and the least common multiple of $x$ and $y$ is $17$ . Find the value of $x+y$ .

Your answer

Show solution

1 · Understandwhat's really being asked

The equation $6x-5y=17$ has infinitely many natural-number solutions. Among them we want the one where $x$ and $y$ have least common multiple $17$ , and then we report $x+y$ .

Givens

$x$ and $y$ are natural numbers.
$6x-5y=17$ .
The least common multiple of $x$ and $y$ is $17$ .

Unknowns

The value of $x+y$ .

Constraints

Both $x$ and $y$ must be natural numbers (positive integers).

2 · Planchoose the strategy

#3 Eliminate Possibilities · also uses: #2 Make a Systematic List#13 Convert to Algebra

Listing solutions of $6x-5y=17$ never ends, so start from the other condition instead. A number always divides any common multiple of itself, so $x$ and $y$ must both divide $17$ . That leaves only four pairs to test against the equation.

3 · Execute3 carry out the plan

1Turn the LCM condition into a short list

#3 Eliminate Possibilities 6.NS.B.4

Each of

x

and

y

divides their least common multiple, which is

17

. The only natural numbers that divide

17

are

1

and

17

.

x \in \{1, 17\}, \quad y \in \{1, 17\}

A prime LCM leaves very few choices for each number.

2Test the four pairs in the equation

#2 Make a Systematic List 6.EE.B.5

Check

6x-5y

for every pair. Only

(17, 17)

gives

17

, and its least common multiple really is

17

.

6 \times 17 - 5 \times 17 = 102 - 85 = 17

The other pairs give

6-5=1

,

6-85=-79

and

102-5=97

, so they fail.

3Add the two values

#13 Convert to Algebra 6.EE.B.5

With

x=17

and

y=17

, the requested sum follows.

x + y = 17 + 17 = 34

The answer is a single number, as expected.

Answer:

34

4 · Reviewdoes it hold up?

Substitute back: $6 \times 17 - 5 \times 17 = 17$ , and the least common multiple of $17$ and $17$ is $17$ . Both conditions hold, so $x+y=34$ .

Another way: List the natural solutions of $6x-5y=17$ directly: each next one has $x$ larger by $5$ , giving $(7, 5)$ , $(12, 11)$ , $(17, 17)$ , $(22, 23)$ , $\ldots$ Their least common multiples are $35$ , $132$ , $17$ , $506$ , $\ldots$ and only $(17, 17)$ has least common multiple $17$ .

Standardsmin grade 6

6.NS.B.4 Find GCF and LCM; distributive property with a common factor — Using that each number divides its least common multiple, so $x$ and $y$ can only be $1$ or $17$.
6.EE.B.5 Solve equations/inequalities by finding values that make them true — Substituting each candidate pair to see which one makes $6x-5y=17$ true.

💡Takeaway. When one condition has endless answers, start from the other condition that allows only a few.

Variant 6 medium

The natural numbers $x$ and $y$ satisfy the linear equation $5x-4y=19$ , and the least common multiple of $x$ and $y$ is $19$ . Find the value of $x+y$ .

Your answer

Show solution

1 · Understandwhat's really being asked

The equation $5x-4y=19$ has infinitely many natural-number solutions. Among them we want the one where $x$ and $y$ have least common multiple $19$ , and then we report $x+y$ .

Givens

$x$ and $y$ are natural numbers.
$5x-4y=19$ .
The least common multiple of $x$ and $y$ is $19$ .

Unknowns

The value of $x+y$ .

Constraints

Both $x$ and $y$ must be natural numbers (positive integers).

2 · Planchoose the strategy

#3 Eliminate Possibilities · also uses: #2 Make a Systematic List#13 Convert to Algebra

Listing solutions of $5x-4y=19$ never ends, so start from the other condition instead. A number always divides any common multiple of itself, so $x$ and $y$ must both divide $19$ . That leaves only four pairs to test against the equation.

3 · Execute3 carry out the plan

1Turn the LCM condition into a short list

#3 Eliminate Possibilities 6.NS.B.4

Each of

x

and

y

divides their least common multiple, which is

19

. The only natural numbers that divide

19

are

1

and

19

.

x \in \{1, 19\}, \quad y \in \{1, 19\}

A prime LCM leaves very few choices for each number.

2Test the four pairs in the equation

#2 Make a Systematic List 6.EE.B.5

Check

5x-4y

for every pair. Only

(19, 19)

gives

19

, and its least common multiple really is

19

.

5 \times 19 - 4 \times 19 = 95 - 76 = 19

The other pairs give

5-4=1

,

5-76=-71

and

95-4=91

, so they fail.

3Add the two values

#13 Convert to Algebra 6.EE.B.5

With

x=19

and

y=19

, the requested sum follows.

x + y = 19 + 19 = 38

The answer is a single number, as expected.

Answer:

38

4 · Reviewdoes it hold up?

Substitute back: $5 \times 19 - 4 \times 19 = 19$ , and the least common multiple of $19$ and $19$ is $19$ . Both conditions hold, so $x+y=38$ .

Another way: List the natural solutions of $5x-4y=19$ directly: each next one has $x$ larger by $4$ , giving $(7, 4)$ , $(11, 9)$ , $(15, 14)$ , $(19, 19)$ , $(23, 24)$ , $\ldots$ Their least common multiples are $28$ , $99$ , $210$ , $19$ , $552$ , $\ldots$ and only $(19, 19)$ has least common multiple $19$ .

Standardsmin grade 6

6.NS.B.4 Find GCF and LCM; distributive property with a common factor — Using that each number divides its least common multiple, so $x$ and $y$ can only be $1$ or $19$.
6.EE.B.5 Solve equations/inequalities by finding values that make them true — Substituting each candidate pair to see which one makes $5x-4y=19$ true.

💡Takeaway. When one condition has endless answers, start from the other condition that allows only a few.

Variant 7 medium

The natural numbers $x$ and $y$ satisfy the linear equation $5x-3y=22$ , and the least common multiple of $x$ and $y$ is $11$ . Find the value of $x+y$ .

Your answer

Show solution

1 · Understandwhat's really being asked

The equation $5x-3y=22$ has infinitely many natural-number solutions. Among them we want the one where $x$ and $y$ have least common multiple $11$ , and then we report $x+y$ .

Givens

$x$ and $y$ are natural numbers.
$5x-3y=22$ .
The least common multiple of $x$ and $y$ is $11$ .

Unknowns

The value of $x+y$ .

Constraints

Both $x$ and $y$ must be natural numbers (positive integers).

2 · Planchoose the strategy

#3 Eliminate Possibilities · also uses: #2 Make a Systematic List#13 Convert to Algebra

Listing solutions of $5x-3y=22$ never ends, so start from the other condition instead. A number always divides any common multiple of itself, so $x$ and $y$ must both divide $11$ . That leaves only four pairs to test against the equation.

3 · Execute3 carry out the plan

1Turn the LCM condition into a short list

#3 Eliminate Possibilities 6.NS.B.4

Each of

x

and

y

divides their least common multiple, which is

11

. The only natural numbers that divide

11

are

1

and

11

.

x \in \{1, 11\}, \quad y \in \{1, 11\}

A prime LCM leaves very few choices for each number.

2Test the four pairs in the equation

#2 Make a Systematic List 6.EE.B.5

Check

5x-3y

for every pair. Only

(11, 11)

gives

22

, and its least common multiple really is

11

.

5 \times 11 - 3 \times 11 = 55 - 33 = 22

The other pairs give

5-3=2

,

5-33=-28

and

55-3=52

, so they fail.

3Add the two values

#13 Convert to Algebra 6.EE.B.5

With

x=11

and

y=11

, the requested sum follows.

x + y = 11 + 11 = 22

The answer is a single number, as expected.

Answer:

22

4 · Reviewdoes it hold up?

Substitute back: $5 \times 11 - 3 \times 11 = 22$ , and the least common multiple of $11$ and $11$ is $11$ . Both conditions hold, so $x+y=22$ .

Another way: List the natural solutions of $5x-3y=22$ directly: each next one has $x$ larger by $3$ , giving $(5, 1)$ , $(8, 6)$ , $(11, 11)$ , $(14, 16)$ , $\ldots$ Their least common multiples are $5$ , $24$ , $11$ , $112$ , $\ldots$ and only $(11, 11)$ has least common multiple $11$ .

Standardsmin grade 6

6.NS.B.4 Find GCF and LCM; distributive property with a common factor — Using that each number divides its least common multiple, so $x$ and $y$ can only be $1$ or $11$.
6.EE.B.5 Solve equations/inequalities by finding values that make them true — Substituting each candidate pair to see which one makes $5x-3y=22$ true.

💡Takeaway. When one condition has endless answers, start from the other condition that allows only a few.

Variant 8 medium

The natural numbers $x$ and $y$ satisfy the linear equation $7x-6y=23$ , and the least common multiple of $x$ and $y$ is $23$ . Find the value of $x+y$ .

Your answer

Show solution

1 · Understandwhat's really being asked

The equation $7x-6y=23$ has infinitely many natural-number solutions. Among them we want the one where $x$ and $y$ have least common multiple $23$ , and then we report $x+y$ .

Givens

$x$ and $y$ are natural numbers.
$7x-6y=23$ .
The least common multiple of $x$ and $y$ is $23$ .

Unknowns

The value of $x+y$ .

Constraints

Both $x$ and $y$ must be natural numbers (positive integers).

2 · Planchoose the strategy

#3 Eliminate Possibilities · also uses: #2 Make a Systematic List#13 Convert to Algebra

Listing solutions of $7x-6y=23$ never ends, so start from the other condition instead. A number always divides any common multiple of itself, so $x$ and $y$ must both divide $23$ . That leaves only four pairs to test against the equation.

3 · Execute3 carry out the plan

1Turn the LCM condition into a short list

#3 Eliminate Possibilities 6.NS.B.4

Each of

x

and

y

divides their least common multiple, which is

23

. The only natural numbers that divide

23

are

1

and

23

.

x \in \{1, 23\}, \quad y \in \{1, 23\}

A prime LCM leaves very few choices for each number.

2Test the four pairs in the equation

#2 Make a Systematic List 6.EE.B.5

Check

7x-6y

for every pair. Only

(23, 23)

gives

23

, and its least common multiple really is

23

.

7 \times 23 - 6 \times 23 = 161 - 138 = 23

The other pairs give

7-6=1

,

7-138=-131

and

161-6=155

, so they fail.

3Add the two values

#13 Convert to Algebra 6.EE.B.5

With

x=23

and

y=23

, the requested sum follows.

x + y = 23 + 23 = 46

The answer is a single number, as expected.

Answer:

46

4 · Reviewdoes it hold up?

Substitute back: $7 \times 23 - 6 \times 23 = 23$ , and the least common multiple of $23$ and $23$ is $23$ . Both conditions hold, so $x+y=46$ .

Another way: List the natural solutions of $7x-6y=23$ directly: each next one has $x$ larger by $6$ , giving $(5, 2)$ , $(11, 9)$ , $(17, 16)$ , $(23, 23)$ , $(29, 30)$ , $\ldots$ Their least common multiples are $10$ , $99$ , $272$ , $23$ , $870$ , $\ldots$ and only $(23, 23)$ has least common multiple $23$ .

Standardsmin grade 6

6.NS.B.4 Find GCF and LCM; distributive property with a common factor — Using that each number divides its least common multiple, so $x$ and $y$ can only be $1$ or $23$.
6.EE.B.5 Solve equations/inequalities by finding values that make them true — Substituting each candidate pair to see which one makes $7x-6y=23$ true.

💡Takeaway. When one condition has endless answers, start from the other condition that allows only a few.

Variant 9 hard

The natural numbers $x$ and $y$ satisfy the linear equation $7x-5y=26$ , and the least common multiple of $x$ and $y$ is $13$ . Find the value of $x+y$ .

Your answer

Show solution

1 · Understandwhat's really being asked

The equation $7x-5y=26$ has infinitely many natural-number solutions. Among them we want the one where $x$ and $y$ have least common multiple $13$ , and then we report $x+y$ .

Givens

$x$ and $y$ are natural numbers.
$7x-5y=26$ .
The least common multiple of $x$ and $y$ is $13$ .

Unknowns

The value of $x+y$ .

Constraints

Both $x$ and $y$ must be natural numbers (positive integers).

2 · Planchoose the strategy

#3 Eliminate Possibilities · also uses: #2 Make a Systematic List#13 Convert to Algebra

Listing solutions of $7x-5y=26$ never ends, so start from the other condition instead. A number always divides any common multiple of itself, so $x$ and $y$ must both divide $13$ . That leaves only four pairs to test against the equation.

3 · Execute3 carry out the plan

1Turn the LCM condition into a short list

#3 Eliminate Possibilities 6.NS.B.4

Each of

x

and

y

divides their least common multiple, which is

13

. The only natural numbers that divide

13

are

1

and

13

.

x \in \{1, 13\}, \quad y \in \{1, 13\}

A prime LCM leaves very few choices for each number.

2Test the four pairs in the equation

#2 Make a Systematic List 6.EE.B.5

Check

7x-5y

for every pair. Only

(13, 13)

gives

26

, and its least common multiple really is

13

.

7 \times 13 - 5 \times 13 = 91 - 65 = 26

The other pairs give

7-5=2

,

7-65=-58

and

91-5=86

, so they fail.

3Add the two values

#13 Convert to Algebra 6.EE.B.5

With

x=13

and

y=13

, the requested sum follows.

x + y = 13 + 13 = 26

The answer is a single number, as expected.

Answer:

26

4 · Reviewdoes it hold up?

Substitute back: $7 \times 13 - 5 \times 13 = 26$ , and the least common multiple of $13$ and $13$ is $13$ . Both conditions hold, so $x+y=26$ .

Another way: List the natural solutions of $7x-5y=26$ directly: each next one has $x$ larger by $5$ , giving $(8, 6)$ , $(13, 13)$ , $(18, 20)$ , $(23, 27)$ , $\ldots$ Their least common multiples are $24$ , $13$ , $180$ , $621$ , $\ldots$ and only $(13, 13)$ has least common multiple $13$ .

Standardsmin grade 6

6.NS.B.4 Find GCF and LCM; distributive property with a common factor — Using that each number divides its least common multiple, so $x$ and $y$ can only be $1$ or $13$.
6.EE.B.5 Solve equations/inequalities by finding values that make them true — Substituting each candidate pair to see which one makes $7x-5y=26$ true.

💡Takeaway. When one condition has endless answers, start from the other condition that allows only a few.

Variant 10 hard

The natural numbers $x$ and $y$ satisfy the linear equation $5x-2y=33$ , and the least common multiple of $x$ and $y$ is $11$ . Find the value of $x+y$ .

Your answer

Show solution

1 · Understandwhat's really being asked

The equation $5x-2y=33$ has infinitely many natural-number solutions. Among them we want the one where $x$ and $y$ have least common multiple $11$ , and then we report $x+y$ .

Givens

$x$ and $y$ are natural numbers.
$5x-2y=33$ .
The least common multiple of $x$ and $y$ is $11$ .

Unknowns

The value of $x+y$ .

Constraints

Both $x$ and $y$ must be natural numbers (positive integers).

2 · Planchoose the strategy

#3 Eliminate Possibilities · also uses: #2 Make a Systematic List#13 Convert to Algebra

Listing solutions of $5x-2y=33$ never ends, so start from the other condition instead. A number always divides any common multiple of itself, so $x$ and $y$ must both divide $11$ . That leaves only four pairs to test against the equation.

3 · Execute3 carry out the plan

1Turn the LCM condition into a short list

#3 Eliminate Possibilities 6.NS.B.4

Each of

x

and

y

divides their least common multiple, which is

11

. The only natural numbers that divide

11

are

1

and

11

.

x \in \{1, 11\}, \quad y \in \{1, 11\}

A prime LCM leaves very few choices for each number.

2Test the four pairs in the equation

#2 Make a Systematic List 6.EE.B.5

Check

5x-2y

for every pair. Only

(11, 11)

gives

33

, and its least common multiple really is

11

.

5 \times 11 - 2 \times 11 = 55 - 22 = 33

The other pairs give

5-2=3

,

5-22=-17

and

55-2=53

, so they fail.

3Add the two values

#13 Convert to Algebra 6.EE.B.5

With

x=11

and

y=11

, the requested sum follows.

x + y = 11 + 11 = 22

The answer is a single number, as expected.

Answer:

22

4 · Reviewdoes it hold up?

Substitute back: $5 \times 11 - 2 \times 11 = 33$ , and the least common multiple of $11$ and $11$ is $11$ . Both conditions hold, so $x+y=22$ .

Another way: List the natural solutions of $5x-2y=33$ directly: each next one has $x$ larger by $2$ , giving $(7, 1)$ , $(9, 6)$ , $(11, 11)$ , $(13, 16)$ , $\ldots$ Their least common multiples are $7$ , $18$ , $11$ , $208$ , $\ldots$ and only $(11, 11)$ has least common multiple $11$ .

Standardsmin grade 6

6.NS.B.4 Find GCF and LCM; distributive property with a common factor — Using that each number divides its least common multiple, so $x$ and $y$ can only be $1$ or $11$.
6.EE.B.5 Solve equations/inequalities by finding values that make them true — Substituting each candidate pair to see which one makes $5x-2y=33$ true.

💡Takeaway. When one condition has endless answers, start from the other condition that allows only a few.

Variant 11 hard

The natural numbers $x$ and $y$ satisfy the linear equation $5x-y=68$ , and the least common multiple of $x$ and $y$ is $17$ . Find the value of $x+y$ .

Your answer

Show solution

1 · Understandwhat's really being asked

The equation $5x-y=68$ has infinitely many natural-number solutions. Among them we want the one where $x$ and $y$ have least common multiple $17$ , and then we report $x+y$ .

Givens

$x$ and $y$ are natural numbers.
$5x-y=68$ .
The least common multiple of $x$ and $y$ is $17$ .

Unknowns

The value of $x+y$ .

Constraints

Both $x$ and $y$ must be natural numbers (positive integers).

2 · Planchoose the strategy

#3 Eliminate Possibilities · also uses: #2 Make a Systematic List#13 Convert to Algebra

Listing solutions of $5x-y=68$ never ends, so start from the other condition instead. A number always divides any common multiple of itself, so $x$ and $y$ must both divide $17$ . That leaves only four pairs to test against the equation.

3 · Execute3 carry out the plan

1Turn the LCM condition into a short list

#3 Eliminate Possibilities 6.NS.B.4

Each of

x

and

y

divides their least common multiple, which is

17

. The only natural numbers that divide

17

are

1

and

17

.

x \in \{1, 17\}, \quad y \in \{1, 17\}

A prime LCM leaves very few choices for each number.

2Test the four pairs in the equation

#2 Make a Systematic List 6.EE.B.5

Check

5x-y

for every pair. Only

(17, 17)

gives

68

, and its least common multiple really is

17

.

5 \times 17 - 1 \times 17 = 85 - 17 = 68

The other pairs give

5-1=4

,

5-17=-12

and

85-1=84

, so they fail.

3Add the two values

#13 Convert to Algebra 6.EE.B.5

With

x=17

and

y=17

, the requested sum follows.

x + y = 17 + 17 = 34

The answer is a single number, as expected.

Answer:

34

4 · Reviewdoes it hold up?

Substitute back: $5 \times 17 - 1 \times 17 = 68$ , and the least common multiple of $17$ and $17$ is $17$ . Both conditions hold, so $x+y=34$ .

Another way: List the natural solutions of $5x-y=68$ directly: each next one has $x$ larger by $1$ , giving $(14, 2)$ , $(15, 7)$ , $(16, 12)$ , $(17, 17)$ , $(18, 22)$ , $\ldots$ Their least common multiples are $14$ , $105$ , $48$ , $17$ , $198$ , $\ldots$ and only $(17, 17)$ has least common multiple $17$ .

Standardsmin grade 6

6.NS.B.4 Find GCF and LCM; distributive property with a common factor — Using that each number divides its least common multiple, so $x$ and $y$ can only be $1$ or $17$.
6.EE.B.5 Solve equations/inequalities by finding values that make them true — Substituting each candidate pair to see which one makes $5x-y=68$ true.

💡Takeaway. When one condition has endless answers, start from the other condition that allows only a few.

Variant 12 hard

The natural numbers $x$ and $y$ satisfy the linear equation $7x-3y=52$ , and the least common multiple of $x$ and $y$ is $13$ . Find the value of $x+y$ .

Your answer

Show solution

1 · Understandwhat's really being asked

The equation $7x-3y=52$ has infinitely many natural-number solutions. Among them we want the one where $x$ and $y$ have least common multiple $13$ , and then we report $x+y$ .

Givens

$x$ and $y$ are natural numbers.
$7x-3y=52$ .
The least common multiple of $x$ and $y$ is $13$ .

Unknowns

The value of $x+y$ .

Constraints

Both $x$ and $y$ must be natural numbers (positive integers).

2 · Planchoose the strategy

#3 Eliminate Possibilities · also uses: #2 Make a Systematic List#13 Convert to Algebra

Listing solutions of $7x-3y=52$ never ends, so start from the other condition instead. A number always divides any common multiple of itself, so $x$ and $y$ must both divide $13$ . That leaves only four pairs to test against the equation.

3 · Execute3 carry out the plan

1Turn the LCM condition into a short list

#3 Eliminate Possibilities 6.NS.B.4

Each of

x

and

y

divides their least common multiple, which is

13

. The only natural numbers that divide

13

are

1

and

13

.

x \in \{1, 13\}, \quad y \in \{1, 13\}

A prime LCM leaves very few choices for each number.

2Test the four pairs in the equation

#2 Make a Systematic List 6.EE.B.5

Check

7x-3y

for every pair. Only

(13, 13)

gives

52

, and its least common multiple really is

13

.

7 \times 13 - 3 \times 13 = 91 - 39 = 52

The other pairs give

7-3=4

,

7-39=-32

and

91-3=88

, so they fail.

3Add the two values

#13 Convert to Algebra 6.EE.B.5

With

x=13

and

y=13

, the requested sum follows.

x + y = 13 + 13 = 26

The answer is a single number, as expected.

Answer:

26

4 · Reviewdoes it hold up?

Substitute back: $7 \times 13 - 3 \times 13 = 52$ , and the least common multiple of $13$ and $13$ is $13$ . Both conditions hold, so $x+y=26$ .

Another way: List the natural solutions of $7x-3y=52$ directly: each next one has $x$ larger by $3$ , giving $(10, 6)$ , $(13, 13)$ , $(16, 20)$ , $(19, 27)$ , $\ldots$ Their least common multiples are $30$ , $13$ , $80$ , $513$ , $\ldots$ and only $(13, 13)$ has least common multiple $13$ .

Standardsmin grade 6

6.NS.B.4 Find GCF and LCM; distributive property with a common factor — Using that each number divides its least common multiple, so $x$ and $y$ can only be $1$ or $13$.
6.EE.B.5 Solve equations/inequalities by finding values that make them true — Substituting each candidate pair to see which one makes $7x-3y=52$ true.

💡Takeaway. When one condition has endless answers, start from the other condition that allows only a few.

Let the LCM shrink the candidate solutions · 12 practice problems

Generated variants — 12

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