Common Denominators: Finding a Common Denominator by Multiplying the Denominators

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: $8, 16, 24, 32, \ldots$

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

The smallest common multiple is $24$.

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

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

The smallest common multiple is $45$.

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: $3, 6, 9, 12, 15, 18, 21, \ldots$

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

The smallest common multiple is $21$.

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: $10, 20, 30, 40, \ldots$

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

The smallest common multiple is $30$.

To find the least common multiple (LCM) of $3$ and $4$, we list the multiples of each number until we find the smallest multiple they have in common.

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

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

The smallest common multiple is $12$.

To find the least common multiple (LCM) of $6$ and $8$, list the multiples of each number until the smallest common multiple appears.

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

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

The smallest common multiple is $24$.

To determine the least common multiple (LCM) of $7$ and $5$, we list the multiples of each number.

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

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

The smallest common multiple is $35$.

To find the least common multiple (LCM) of $9$ and $6$, list the multiples of each number until the smallest common multiple appears.

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

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

The smallest common multiple is $18$.

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: $4, 8, 12, 16, \ldots$

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

The smallest common multiple is $12$.

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

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

The smallest common multiple is $10$.

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

Let's try to find the lowest common denominator between 5 and 3

To find the lowest common denominator, we need to find a number that is divisible by both 5 and 3

In this case, the common denominator is 15

Now we'll multiply each fraction by the appropriate number to reach the denominator 15

We'll multiply the first fraction by 3

We'll multiply the second fraction by 5

$\frac{2\times3}{5\times3}+\frac{1\times5}{3\times5}=\frac{6}{15}+\frac{5}{15}$

Now we'll combine and get:

$\frac{6+5}{15}=\frac{11}{15}$

Let's try to find the lowest common denominator between 6 and 7

To find the lowest common denominator, we need to find a number that is divisible by both 6 and 7

In this case, the common denominator is 42

Now we'll multiply each fraction by the appropriate number to reach the denominator 42

We'll multiply the first fraction by 7

We'll multiply the second fraction by 6

$\frac{2\times7}{6\times7}+\frac{3\times6}{7\times6}=\frac{14}{42}+\frac{18}{42}$

Now we'll combine and get:

$\frac{14+18}{42}=\frac{32}{42}$