Examples with solutions for Common Denominators: Finding a Common Denominator by Multiplying the Denominators

Exercise #1

What is the least common multiple (LCM) of the numbers 3 and 7?

3   7 \boxed 3~~~\boxed 7

Step-by-Step Solution

To find the least common multiple (LCM) of the numbers 3 and 7, we will list the multiples of each number and find the smallest multiple they have in common.

Multiples of 3: 3,6,9,12,15,18,21, 3, 6, 9, 12, 15, 18, 21, \ldots

Multiples of 7: 7,14,21,28, 7, 14, 21, 28, \ldots

The smallest common multiple is 21 21 .

Answer

21

Exercise #2

You have two numbers, 4 and 6. What is their least common multiple?

4   6 \boxed 4~~~\boxed 6

Step-by-Step Solution

To find the least common multiple (LCM) of the numbers 4 and 6, list the multiples of each number and find the smallest multiple they have in common.

Multiples of 4: 4,8,12,16, 4, 8, 12, 16, \ldots

Multiples of 6: 6,12,18,24, 6, 12, 18, 24, \ldots

The smallest common multiple is 12 12 .

Answer

12

Exercise #3

Find the least common multiple of 5 and 9.

5   9 \boxed 5~~~\boxed 9

Step-by-Step Solution

To find the least common multiple (LCM) of 5 and 9, list the multiples of each number and find the smallest multiple they have in common.

Multiples of 5: 5,10,15,20,25,30,35,40,45, 5, 10, 15, 20, 25, 30, 35, 40, 45, \ldots

Multiples of 9: 9,18,27,36,45, 9, 18, 27, 36, 45, \ldots

The smallest common multiple is 45 45 .

Answer

45

Exercise #4

Determine the least common multiple of 8 and 12.

8   12 \boxed 8~~~\boxed{ 12}

Step-by-Step Solution

To find the least common multiple (LCM) of the numbers 8 and 12, we will list the multiples of each number and find the smallest multiple they have in common.

Multiples of 8: 8,16,24,32, 8, 16, 24, 32, \ldots

Multiples of 12: 12,24,36,48, 12, 24, 36, 48, \ldots

The smallest common multiple is 24 24 .

Answer

24

Exercise #5

What is the least common multiple of the numbers 10 and 15?

10   15 \boxed {10}~~~\boxed {15 }

Step-by-Step Solution

To find the least common multiple (LCM) of 10 and 15, list the multiples of each number and find the smallest multiple they have in common.

Multiples of 10: 10,20,30,40, 10, 20, 30, 40, \ldots

Multiples of 15: 15,30,45,60, 15, 30, 45, 60, \ldots

The smallest common multiple is 30 30 .

Answer

30

Exercise #6

You have a pair of denominators, what is their least common multiple?

3   4 \boxed 3~~~\boxed 4

Step-by-Step Solution

To find the least common multiple (LCM) of 33 and 44, we list the multiples of each number until we find the smallest multiple they have in common.

Multiples of 33: 3,6,9,12,15,3, 6, 9, 12, 15, \ldots

Multiples of 44: 4,8,12,16,20,4, 8, 12, 16, 20, \ldots

The smallest common multiple is 1212.

Answer

12

Exercise #7

You have a pair of denominators, what is their least common multiple?

6   8 \boxed 6~~~\boxed 8

Step-by-Step Solution

To find the least common multiple (LCM) of 66 and 88, list the multiples of each number until the smallest common multiple appears.

Multiples of 66: 6,12,18,24,30,6, 12, 18, 24, 30, \ldots

Multiples of 88: 8,16,24,32,40,8, 16, 24, 32, 40, \ldots

The smallest common multiple is 2424.

Answer

24

Exercise #8

You have a pair of denominators, what is their least common multiple?

7   5 \boxed 7~~~\boxed 5

Step-by-Step Solution

To determine the least common multiple (LCM) of 77 and 55, we list the multiples of each number.

Multiples of 77: 7,14,21,28,35,7, 14, 21, 28, 35, \ldots

Multiples of 55: 5,10,15,20,25,30,35,5, 10, 15, 20, 25, 30, 35, \ldots

The smallest common multiple is 3535.

Answer

35

Exercise #9

You have a pair of denominators, what is their least common multiple?

9   6 \boxed 9~~~\boxed 6

Step-by-Step Solution

To find the least common multiple (LCM) of 99 and 66, list the multiples of each number until the smallest common multiple appears.

Multiples of 99: 9,18,27,36,9, 18, 27, 36, \ldots

Multiples of 66: 6,12,18,24,30,6, 12, 18, 24, 30, \ldots

The smallest common multiple is 1818.

Answer

18

Exercise #10

What is the least common multiple of these two numbers?

2   5 \boxed{2}~~~\boxed{5}

Step-by-Step Solution

To find the least common multiple (LCM) of 2 2 and 5 5 , we list the multiples of each number:

  • Multiples of 2 2 are 2,4,6,8,10, 2, 4, 6, 8, 10, \ldots
  • Multiples of 5 5 are 5,10,15, 5, 10, 15, \ldots

The smallest common multiple is 10 10 .

Answer

10

Exercise #11

Given several denominators, what is their least common multiple?

2   5   7 \boxed{2}~~~\boxed{5} ~~~\boxed{7}

Step-by-Step Solution

To find the least common multiple (LCM) of 2 2 , 5 5 , and 7 7 , start with the prime factorizations:

2 2 , 5 5 , and 7 7 , as they all are primes.

The LCM is simply their product: 2×5×7=70 2 \, \times \, 5 \, \times \, 7 = 70 .

Answer

70

Exercise #12

Determine the least common multiple (LCM) of the following numerators:

3   7   5 \boxed{3} ~~~ \boxed{7} ~~~ \boxed{5}

Step-by-Step Solution

To find the least common multiple (LCM) of the numbers 3, 7, and 5, we use the prime factorization method:

Prime factors of each number:

  • 3: 31 3^1
  • 7: 71 7^1
  • 5: 51 5^1

The LCM is the product of the highest powers of all prime factors:

31×71×51=3×7×5=105 3^1 \times 7^1 \times 5^1 = 3 \times 7 \times 5 = 105

Answer

105

Exercise #13

Given several denominators, what is their least common multiple?

573 \boxed{5} \boxed{7} \boxed{3}

Step-by-Step Solution

The least common multiple (LCM) of 5,7, and 35, 7, \text{ and } 3 is the smallest positive integer that is divisible by each of these numbers.

First, calculate the LCM by multiplying the numbers, as they are all prime:

LCM is 5×7×3=1055 \times 7 \times 3 = 105.

So the least common multiple is 105105.

Answer

105

Exercise #14

Given several denominators, what is their least common multiple?

91113 \boxed{9} \boxed{11} \boxed{13}

Step-by-Step Solution

The least common multiple (LCM) of 9,11, and 139, 11, \text{ and } 13 is the smallest positive integer that is divisible by each of these numbers.

Since there are no common factors other than 1, the LCM is simply the product of these numbers:

9×11×139 \times 11 \times 13 equals 1287.

The LCM is 12871287.

Answer

1287

Exercise #15

Given three denominators, what is their least common multiple?

3   7   2 \boxed{3}~~~\boxed{7}~~~\boxed{2}

Step-by-Step Solution

To find the least common multiple (LCM) of the denominators 3, 7, and 2, we find the smallest positive integer that is divisible by all three numbers. The prime factors are:

3=33 = 3

7=77 = 7

2=22 = 2

Since all numbers are primes, the least common multiple is simply their product:

3×7×2=423 \times 7 \times 2 = 42

Answer

42

Exercise #16

Given several denominators, what is their least common multiple?

2   7   9 \boxed2~~~\boxed7 ~~~\boxed9

Step-by-Step Solution

To find the least common multiple (LCM) of the denominators 2,7,92, 7, 9, we need to consider each prime factor of these numbers at their highest power:

  • 22: prime itself

  • 77: prime itself

  • 9=329 = 3^2

Therefore, the LCM is:

2×7×32=1262 \times 7 \times 3^2 = 126

So, the least common multiple of 2,7,92, 7, 9 is 126126.

Answer

126

Exercise #17

What is the least common multiple of the following numbers?

7   11   13 \boxed{7} ~~~ \boxed{11} ~~~ \boxed{13}

Step-by-Step Solution

To find the least common multiple (LCM) of the numbers 7, 11, and 13, we first recognize that all these numbers are prime. The LCM is given by multiplying these numbers together.

The formula for the LCM of three numbers a,b,c a, b, c is:

LCM(a,b,c)=a×b×c \text{LCM}(a, b, c) = a \times b \times c

Substituting into the formula gives:

LCM(7,11,13)=7×11×13 \text{LCM}(7, 11, 13) = 7 \times 11 \times 13

Calculating the product:

7×11=77 7 \times 11 = 77

77×13=1001 77 \times 13 = 1001

So, the least common multiple of 7, 11, and 13 is 1001.

Answer

1001

Exercise #18

Given several denominators, what is their least common multiple?

6   8   9 \boxed{6}~~~\boxed{8} ~~~\boxed{9}

Step-by-Step Solution

To find the least common multiple (LCM) of 6 6 , 8 8 and 9 9 , we start by finding the prime factors of each number:

6=2×3 6 = 2 \, \times \, 3

8=23 8 = 2^3

9=32 9 = 3^2

The LCM is found by taking the highest power of each prime that appears in these factorizations:

23 2^3 (from 8), and 32 3^2 (from 9).

The LCM is 23×32=8×9=72 2^3 \, \times \, 3^2 = 8 \, \times \, 9 = 72 .

Answer

72

Exercise #19

Solve the following exercise:

14+19= \frac{1}{4}+\frac{1}{9}=

Video Solution

Step-by-Step Solution

Let's try to find the lowest common denominator between 4 and 9

To find the lowest common denominator, we need to find a number that is divisible by both 4 and 9

In this case, the common denominator is 36

Now we'll multiply each fraction by the appropriate number to reach the denominator 36

We'll multiply the first fraction by 9

We'll multiply the second fraction by 4

1×94×9+1×49×4=936+436 \frac{1\times9}{4\times9}+\frac{1\times4}{9\times4}=\frac{9}{36}+\frac{4}{36}

Now we'll combine and get:

9+436=1336 \frac{9+4}{36}=\frac{13}{36}

Answer

1336 \frac{13}{36}

Exercise #20

Solve the following exercise:

17+13= \frac{1}{7}+\frac{1}{3}=

Video Solution

Step-by-Step Solution

Let's try to find the lowest common denominator between 7 and 3

To find the lowest common denominator, we need to find a number that is divisible by both 7 and 3

In this case, the common denominator is 21

Now we'll multiply each fraction by the appropriate number to reach the denominator 21

We'll multiply the first fraction by 3

We'll multiply the second fraction by 7

1×37×3+1×73×7=321+721 \frac{1\times3}{7\times3}+\frac{1\times7}{3\times7}=\frac{3}{21}+\frac{7}{21}

Now we'll combine and get:

3+721=1021 \frac{3+7}{21}=\frac{10}{21}

Answer

1021 \frac{10}{21}