Common Denominators: The common denominator is smaller than the product of the denominators

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

$8 = 2^3$

$5 = 5$

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

$2^3 \times 5 = 8 \times 5 = 40$

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

$3 = 3$

$9 = 3^2$

$12 = 2^2 \, \times \, 3$

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

Max of $2$ is $2^2$ and of $3$ is $3^2$.

Thus, LCM is $2^2 \, \times \, 3^2 = 4 \, \times \, 9 = 36$.

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

$8 = 2^3$

$16 = 2^4$

$20 = 2^2 \, \times \, 5$

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

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

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

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

$18 = 2 \, \times \, 3^2$

$24 = 2^3 \, \times \, 3$

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

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

$2^3$ from 24, $3^2$ from 18, and $5^1$ from 30.

The LCM is $2^3 \, \times \, 3^2 \, \times \, 5 = 8 \, \times \, 9 \, \times \, 5 = 360$.

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

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

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

Prime factors of 6: $6 = 2^1 \times 3^1$

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

$2^2 \times 3^2 = 4 \times 9 = 36$

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

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

Prime factors of 14: $14 = 2^1 \times 7^1$

Prime factors of 20: $20 = 2^2 \times 5^1$

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

$2^3 \times 7^1 \times 5^1 = 8 \times 35 = 280$

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

$10 = 2 \, \times \, 5$

$15 = 3 \, \times \, 5$

$25 = 5^2$

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

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

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

First, list the multiples of each number:

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

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

The smallest common multiple is $24$.

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

List the multiples for reference:

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

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

The smallest common multiple is $60$.

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

Using the prime factors, we find:

• The prime factors of $14$ are 2 and 7.
• The prime factors of $15$ are 3 and 5.
• The prime factors of $3$ is 3.

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

Therefore, the least common multiple is $105$.

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

$4 = 2^2$

$6 = 2 \times 3$

$9 = 3^2$

$3 = 3$

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

$2^2 \times 3^2 = 4 \times 9 = 36$

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

$5 = 5$

$4 = 2^2$

$6 = 2 \times 3$

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

$5 \times 2^2 \times 3 = 60$

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

$3 = 3$

$5 = 5$

$12 = 2^2 \times 3$

$15 = 3 \times 5$

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

$2^2 \times 3 \times 5 = 60$

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

• $6 = 2 \times 3$

• $8 = 2^3$

• $10 = 2 \times 5$

Therefore, the LCM is:

$2^3 \times 3 \times 5 = 120$

So, the least common multiple of $6, 8, 10$ is $120$.

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

• $5$: prime itself

• $6 = 2 \times 3$

• $15 = 3 \times 5$

Therefore, the LCM is:

$2 \times 3 \times 5 = 30$

So, the least common multiple of $5, 6, 15$ is $30$.

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

$\frac{1\times3}{4\times3}+\frac{3\times2}{6\times2}=\frac{3}{12}+\frac{6}{12}$

Finally we'll combine and obtain the following:

$\frac{6+3}{12}=\frac{9}{12}$

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

$\frac{4\times5}{8\times5}+\frac{4\times4}{10\times4}=\frac{20}{40}+\frac{16}{40}$

Now let's calculate:

$\frac{20+16}{40}=\frac{36}{40}$

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

$\frac{3\times3}{6\times3}+\frac{3\times2}{9\times2}=\frac{9}{18}+\frac{6}{18}$

Finally we'll combine and obtain the following:

$\frac{9+6}{18}=\frac{15}{18}$

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

$\frac{2\times3}{4\times3}+\frac{1\times2}{6\times2}=\frac{6}{12}+\frac{2}{12}$

Finally let's combine to obtain the following.

$\frac{6+2}{12}=\frac{8}{12}$

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

$\frac{4\times3}{8\times3}+\frac{5\times2}{12\times2}=\frac{12}{24}+\frac{10}{24}$

Now let's add:

$\frac{12+10}{24}=\frac{22}{24}$