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Least Common Multiple
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When we learned adding and subtracting fractions, we learned that the number on the bottom, the denominator, has to be the same. In order to get a common denominator, we first need to find the Least Common Multiple (LCM). The LCM of two numbers is the smallest number that both numbers divide evenly. The video below will explain two methods of finding LCMs:

Least Common Multiple (LCM)

### The two ways to find the LCM:

1. Skip Counting
• List the multiples of each of the numbers given and find the smallest number in both lists.
2. Prime Factorization
• Find all the prime factors of each number given.
• Create a new number that contains all the prime factors of each number. Remember to include multiples if there are multiples of the same factor in either prime factorization. Example: $$9 = 3 \times 3$$ and $$15 = 3 \times 5$$, since 9 has two 3s and 15 has only one 3 in its factorization, the combined list will need two 3s. The LCM of 9 and 15 is $$3 \times 3 \times 5 = 45$$

### Practice Problems

Find the least common multiple for the following pairs:
1. 5 and 6 (
Solution
Solution: 30
Details: To find the least common multiple (LCM), we will start by finding the prime factors of both 5 and 6.

Since 5 is prime, the prime factorization of 5 is just itself. Since 2 and 3 are both prime, we can write the prime factorization of 6 as $$2 \times 3$$.

We have the following:
$$5 = 5$$
$$6 = 2 \times 3$$

Our LCM will need to have the prime factors of both 5 and 6, so it needs to include 2, 3, and 5. $$LCM = 2 \times 3 \times 5 = 30$$
)
2. 4 and 12 (
Solution
Solution: 12
Details: To find the least common multiple (LCM), we will start by finding the prime factors of both 4 and 12. Since 2 is prime, we can write the prime factorization of 4 as $$2 \times 2$$. Since 2 and 3 are both prime, we can write the prime factorization of 12 as $$2 \times 2 \times 3$$.

We have the following:
$$4 = 2 \times 2$$
$$12 = 2 \times 2 \times 3$$

Our LCM will need to have the prime factors of both 4 and 12. Since the prime factorization of 4 is $$2 \times 2$$ and the prime factorization of 12 has $$2 \times 2$$ in it, we will only need to include $$2 \times 2$$ once. $$LCM = 2 \times 2 \times 3 = 12$$
)
3. 6 and 10 (
Video Solution
Solution: 30

(Video Source | Transcript)

Detail: To find the least common multiple (LCM), we will start by finding the prime factors of both 6 and 10. Since 2 and 3 are both prime, we can write the prime factorization of 6 as $$2 \times 3$$. Since 2 and 5 are both prime, we can write the prime factorization of 10 as $$2 \times 5$$.

We have the following:
$$6 = 2 \times 3$$
$$10 = 2 \times 5$$

Our LCM will need to have the prime factors of both 6 and 10. Since the prime factorization of both 6 and 10 include a 2, we will only need to include 2 once. $$LCM = 2 \times 3 \times 5 = 30$$
)
4. 4 and 14 (
Solution
Solution: 28
Details: To find the least common multiple (LCM), we will start by finding the prime factors of both 4 and 14. Since 2 is prime, we can write the prime factorization of 4 as $$2 \times 2$$. Since 2 and 7 are both prime, we can write the prime factorization of 14 as $$2 \times 7$$.

We have the following:
$$4 = 2 \times 2$$
$$14 = 2 \times 7$$

Our LCM will need to have the prime factors of both 4 and 14. Since the prime factorization of 4 is $$2 \times 2$$, when we include the prime factors of 14 we already have at least one 2, so we do not need to include another 2. $$LCM = 2 \times 2 \times 7 = 28$$
)
5. 7 and 9 (
Video Solution
Solution: 63

(Video Source | Transcript)

Detail: To find the least common multiple (LCM), we will start by finding the prime factors of both 7 and 9.

Since 7 is prime, the prime factorization of 7 is just itself. Since 3 is prime, we can write the prime factorization of 9 as $$3 \times 3$$.

We have the following:
$$7 = 7$$
$$9 = 3 \times 3$$

Our LCM will need to have the prime factors of both 7 and 9, so we will include both 7 and $$3 \times 3$$. $$LCM = 7 × 3 × 3 = 63$$
)
6. 7 and 5 (
Solution
Solution: 35
Details: To find the least common multiple (LCM), we will start by finding the prime factors of both 7 and 5. Since 7 and 5 are both prime, the prime factorization of each is just itself.

We have the following:
$$5 = 5$$
$$7 = 7$$

Our LCM will need to have the prime factors of both 7 and 5, so we will include both 7 and 5. $$LCM = 7 \times 5 = 35$$
)

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