Examples with solutions for Common Denominators: The common denominator is smaller than the product of the denominators

Exercise #1

Given two denominators, what is their least common multiple?

8   5 \boxed{8}~~~\boxed{5}

Step-by-Step Solution

To find the least common multiple (LCM) of 8 and 5, identify the prime factors:

8=238 = 2^3

5=55 = 5

The LCM is the product of the highest power of each prime:

23×5=8×5=402^3 \times 5 = 8 \times 5 = 40

Answer

40

Exercise #2

Identify the least common multiple of these denominators:

3   9   12 \boxed{3}~~~\boxed{9} ~~~\boxed{12}

Step-by-Step Solution

To find the least common multiple (LCM) of 3 3 , 9 9 , and 12 12 , begin by finding their prime factorizations:

3=3 3 = 3

9=32 9 = 3^2

12=22×3 12 = 2^2 \, \times \, 3

The LCM is calculated by taking the highest power of each prime present:

Max of 2 2 is 22 2^2 and of 3 3 is 32 3^2 .

Thus, LCM is 22×32=4×9=36 2^2 \, \times \, 3^2 = 4 \, \times \, 9 = 36 .

Answer

36

Exercise #3

What is the least common multiple of these denominators?

8   16   20 \boxed{8}~~~\boxed{16} ~~~\boxed{20}

Step-by-Step Solution

To find the least common multiple (LCM) of 8 8 , 16 16 , and 20 20 , find their prime factorizations:

8=23 8 = 2^3

16=24 16 = 2^4

20=22×5 20 = 2^2 \, \times \, 5

The LCM is obtained by taking the highest power of each prime number:

24 2^4 from 16 and 5 5 from 20.

The LCM is 24×5=16×5=80 2^4 \, \times \, 5 = 16 \, \times \, 5 = 80 .

Answer

80

Exercise #4

Among these numbers, what is the least common multiple?

18   24   30 \boxed{18}~~~\boxed{24} ~~~\boxed{30}

Step-by-Step Solution

To find the least common multiple (LCM) of 18 18 , 24 24 , and 30 30 , first find their prime factorizations:

18=2×32 18 = 2 \, \times \, 3^2

24=23×3 24 = 2^3 \, \times \, 3

30=2×3×5 30 = 2 \, \times \, 3 \, \times \, 5

The LCM is obtained by using the highest power of each prime:

23 2^3 from 24, 32 3^2 from 18, and 51 5^1 from 30.

The LCM is 23×32×5=8×9×5=360 2^3 \, \times \, 3^2 \, \times \, 5 = 8 \, \times \, 9 \, \times \, 5 = 360 .

Answer

360

Exercise #5

Calculate the least common multiple (LCM) for these numbers:

9   4   6 \boxed{9} ~~~ \boxed{4} ~~~ \boxed{6}

Step-by-Step Solution

To find the least common multiple (LCM) of the numbers 9, 4, and 6, use their prime factors:

Prime factors of 9: 9=32 9 = 3^2

Prime factors of 4: 4=22 4 = 2^2

Prime factors of 6: 6=21×31 6 = 2^1 \times 3^1

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

22×32=4×9=36 2^2 \times 3^2 = 4 \times 9 = 36

Answer

36

Exercise #6

Find the least common multiple (LCM) of these numbers:

8   14   20 \boxed{8} ~~~ \boxed{14} ~~~ \boxed{20}

Step-by-Step Solution

To find the least common multiple (LCM) of 8, 14, and 20, determine their prime factorization:

Prime factors of 8: 8=23 8 = 2^3

Prime factors of 14: 14=21×71 14 = 2^1 \times 7^1

Prime factors of 20: 20=22×51 20 = 2^2 \times 5^1

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

23×71×51=8×35=280 2^3 \times 7^1 \times 5^1 = 8 \times 35 = 280

Answer

280

Exercise #7

Given several denominators, what is their least common multiple?

10   15   25 \boxed{10}~~~\boxed{15} ~~~\boxed{25}

Step-by-Step Solution

To find the least common multiple (LCM) of 10 10 , 15 15 , and 25 25 , we first find their prime factorizations:

10=2×5 10 = 2 \, \times \, 5

15=3×5 15 = 3 \, \times \, 5

25=52 25 = 5^2

The LCM is found by multiplying the highest powers of every prime number: 21 2^1 , 31 3^1 , and 52 5^2 .

LCM = 2×3×25=6×25=150 2 \, \times \, 3 \, \times \, 25 = 6 \, \times \, 25 = 150 .

Answer

150

Exercise #8

Given several denominators, what is their least common multiple?

346 \boxed{3} \boxed{4} \boxed{6}

Step-by-Step Solution

The least common multiple (LCM) of 3,4, and 63, 4, \text{ and } 6 is the smallest positive integer that is divisible by each of these numbers.

First, list the multiples of each number:

  • Multiples of 33: 3, 6, 9, 12, 15, 18, 21, 24, ...
  • Multiples of 44: 4, 8, 12, 16, 20, 24, ...
  • Multiples of 66: 6, 12, 18, 24, ...

The common multiples of 3,4, and 63, 4, \text{ and } 6 are 12, 24, ...

The smallest common multiple is 2424.

Answer

24

Exercise #9

Given several denominators, what is their least common multiple?

81012 \boxed{8} \boxed{10} \boxed{12}

Step-by-Step Solution

The least common multiple (LCM) of 8,10, and 128, 10, \text{ and } 12 is the smallest positive integer that is divisible by each of these numbers.

List the multiples for reference:

  • Multiples of 88: 8, 16, 24, 32, 40, 48, 56, 64, ...
  • Multiples of 1010: 10, 20, 30, 40, 50, 60, ...
  • Multiples of 1212: 12, 24, 36, 48, 60, ...

The common multiples of 8,10, and 128, 10, \text{ and } 12 are 60, 120, ...

The smallest common multiple is 6060.

Answer

60

Exercise #10

Given several denominators, what is their least common multiple?

14153 \boxed{14} \boxed{15} \boxed{3}

Step-by-Step Solution

The least common multiple (LCM) of 14,15, and 314, 15, \text{ and } 3 is the smallest positive integer that is divisible by each of these numbers.

Using the prime factors, we find:

  • The prime factors of 1414 are 2 and 7.
  • The prime factors of 1515 are 3 and 5.
  • The prime factors of 33 is 3.

The LCM will be 2×3×5×7=2102 \times 3 \times 5 \times 7 = 210.

Therefore, the least common multiple is 105105.

Answer

105

Exercise #11

Given four denominators, what is their least common multiple?

4   6   9   3 \boxed{4}~~~\boxed{6}~~~\boxed{9}~~~\boxed{3}

Step-by-Step Solution

To find the least common multiple (LCM) of 4, 6, 9, and 3, we start by identifying the prime factors:

4=224 = 2^2

6=2×36 = 2 \times 3

9=329 = 3^2

3=33 = 3

The LCM will be found by taking the highest power of each prime present:

22×32=4×9=362^2 \times 3^2 = 4 \times 9 = 36

Answer

36

Exercise #12

Given three denominators, what is their least common multiple?

5   4   6 \boxed{5}~~~\boxed{4}~~~\boxed{6}

Step-by-Step Solution

To find the least common multiple (LCM) of 5, 4, and 6, identify the prime factors:

5=55 = 5

4=224 = 2^2

6=2×36 = 2 \times 3

The LCM is obtained by taking the highest power of each prime:

5×22×3=605 \times 2^2 \times 3 = 60

Answer

60

Exercise #13

Given four denominators, what is their least common multiple?

3   5   12   15 \boxed{3}~~~\boxed{5}~~~\boxed{12}~~~\boxed{15}

Step-by-Step Solution

To find the least common multiple (LCM) of 3, 5, 12, and 15, we identify their prime factors:

3=33 = 3

5=55 = 5

12=22×312 = 2^2 \times 3

15=3×515 = 3 \times 5

The LCM is obtained by taking the highest power of each prime:

22×3×5=602^2 \times 3 \times 5 = 60

Answer

60

Exercise #14

Given several denominators, what is their least common multiple?

6   8   10 \boxed6~~~\boxed8 ~~~\boxed{10 }

Step-by-Step Solution

To find the least common multiple (LCM) of the denominators 6,8,106, 8, 10, we need to consider each prime factor of these numbers at their highest power:

  • 6=2×36 = 2 \times 3

  • 8=238 = 2^3

  • 10=2×510 = 2 \times 5

Therefore, the LCM is:

23×3×5=1202^3 \times 3 \times 5 = 120

So, the least common multiple of 6,8,106, 8, 10 is 120120.

Answer

120

Exercise #15

Given several denominators, what is their least common multiple?

5   6   15 \boxed5~~~\boxed6 ~~~\boxed{15}

Step-by-Step Solution

To find the least common multiple (LCM) of the denominators 5,6,155, 6, 15, we need to consider each prime factor of these numbers at their highest power:

  • 55: prime itself

  • 6=2×36 = 2 \times 3

  • 15=3×515 = 3 \times 5

Therefore, the LCM is:

2×3×5=302 \times 3 \times 5 = 30

So, the least common multiple of 5,6,155, 6, 15 is 3030.

Answer

30

Exercise #16

Solve the following equation:

14+36= \frac{1}{4}+\frac{3}{6}=

Video Solution

Step-by-Step Solution

We must first identify the lowest common denominator between 4 and 6.

In order to determine the lowest common denominator, we need to find a number that is divisible by both 4 and 6.

In this case, the common denominator is 12.

We will then proceed to multiply each fraction by the appropriate number to reach the denominator 12

We'll multiply the first fraction by 3

We'll multiply the second fraction by 2

1×34×3+3×26×2=312+612 \frac{1\times3}{4\times3}+\frac{3\times2}{6\times2}=\frac{3}{12}+\frac{6}{12}

Finally we'll combine and obtain the following:

6+312=912 \frac{6+3}{12}=\frac{9}{12}

Answer

912 \frac{9}{12}

Exercise #17

48+410= \frac{4}{8}+\frac{4}{10}=

Video Solution

Step-by-Step Solution

Let's try to find the lowest common multiple between 8 and 10

To find the lowest common multiple, we need to find a number that is divisible by both 8 and 10

In this case, the lowest common multiple is 40

Now, let's multiply each number in the appropriate multiples to reach the number 40

We will multiply the first number by 5

We will multiply the second number by 4

4×58×5+4×410×4=2040+1640 \frac{4\times5}{8\times5}+\frac{4\times4}{10\times4}=\frac{20}{40}+\frac{16}{40}

Now let's calculate:

20+1640=3640 \frac{20+16}{40}=\frac{36}{40}

Answer

3640 \frac{36}{40}

Exercise #18

Solve the following equation:

36+39= \frac{3}{6}+\frac{3}{9}=

Video Solution

Step-by-Step Solution

We must first identify the lowest common denominator between 6 and 9.

In order to determine the lowest common denominator, we need to find a number that is divisible by both 6 and 9.

In this case, the common denominator is 18.

We will then proceed to multiply each fraction by the appropriate number to reach the denominator 18.

We'll multiply the first fraction by 3

We'll multiply the second fraction by 2

3×36×3+3×29×2=918+618 \frac{3\times3}{6\times3}+\frac{3\times2}{9\times2}=\frac{9}{18}+\frac{6}{18}

Finally we'll combine and obtain the following:

9+618=1518 \frac{9+6}{18}=\frac{15}{18}

Answer

1518 \frac{15}{18}

Exercise #19

Solve the following equation:

24+16= \frac{2}{4}+\frac{1}{6}=

Video Solution

Step-by-Step Solution

Let's begin by identifying the lowest common denominator between 4 and 6.

In order to determine the lowest common denominator, we need to find a number that is divisible by both 4 and 6.

In this case, the common denominator is 12.

We'll proceed to multiply each fraction by the appropriate number to reach the denominator 12.

We'll multiply the first fraction by 3

We'll multiply the second fraction by 2

2×34×3+1×26×2=612+212 \frac{2\times3}{4\times3}+\frac{1\times2}{6\times2}=\frac{6}{12}+\frac{2}{12}

Finally let's combine to obtain the following.

6+212=812 \frac{6+2}{12}=\frac{8}{12}

Answer

812 \frac{8}{12}

Exercise #20

Solve the following equation:

48+512= \frac{4}{8}+\frac{5}{12}=

Video Solution

Step-by-Step Solution

Let's first identify the lowest common denominator between 8 and 12.

In order to identify the lowest common denominator, we need to find a number that is divisible by both 8 and 12.

In this case, the common denominator is 24.

Let's proceed to multiply each fraction by the appropriate number to reach the denominator 24.

We'll multiply the first fraction by 3

We'll multiply the second fraction by 2

4×38×3+5×212×2=1224+1024 \frac{4\times3}{8\times3}+\frac{5\times2}{12\times2}=\frac{12}{24}+\frac{10}{24}

Now let's add:

12+1024=2224 \frac{12+10}{24}=\frac{22}{24}

Answer

2224 \frac{22}{24}