Common Denominators: More than two fractions

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: $3^1$
• 7: $7^1$
• 5: $5^1$

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

$3^1 \times 7^1 \times 5^1 = 3 \times 7 \times 5 = 105$

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

• $2$: prime itself

• $7$: prime itself

• $9 = 3^2$

Therefore, the LCM is:

$2 \times 7 \times 3^2 = 126$

So, the least common multiple of $2, 7, 9$ is $126$.

The least common multiple (LCM) of $5, 7, \text{ and } 3$ 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 $5 \times 7 \times 3 = 105$.

So the least common multiple is $105$.

The least common multiple (LCM) of $9, 11, \text{ and } 13$ 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:

$9 \times 11 \times 13$ equals 1287.

The LCM is $1287$.

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:

$3 = 3$

$7 = 7$

$2 = 2$

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

$3 \times 7 \times 2 = 42$

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 $a, b, c$ is:

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

Substituting into the formula gives:

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

Calculating the product:

$7 \times 11 = 77$

$77 \times 13 = 1001$

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

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

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

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

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

$6 = 2 \, \times \, 3$

$8 = 2^3$

$9 = 3^2$

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

$2^3$ (from 8), and $3^2$ (from 9).

The LCM is $2^3 \, \times \, 3^2 = 8 \, \times \, 9 = 72$.

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 $5$, $10$, and $20$, find their prime factorizations:

$5 = 5$

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

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

$\times$The LCM is determined by selecting the greatest power of each prime number:

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

The LCM is $2^2 \, \times \, 5 = 4 \, \times \, 5 = 20$.

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:

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