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### What is the Least Common Multiple?

The Least Common Multiple (LCM) is the smallest positive integer that is divisible by two or more given numbers without leaving any remainder. It represents the least common multiple of the numbers and is often used in various mathematical operations, such as simplifying fractions, adding and subtracting fractions with different denominators, and solving algebraic equations.

Here are two examples of finding the LCM of sets of numbers:

Example 1: Find the LCM of 6 and 8

To find the LCM of 6 and 8, you can list the multiples of each number and identify the smallest number that appears in both lists.

Multiples of 6: 6, 12, 18, 24, 30, 36, …

Multiples of 8: 8, 16, 24, 32, 40, …

The smallest number that appears in both lists is 24. So, the LCM of 6 and 8 is 24.

Example 2: Find the LCM of 15 and 20

To find the LCM of 15 and 20, you can again list the multiples of each number and identify the smallest number that appears in both lists.

Multiples of 15: 15, 30, 45, 60, 75, …

Multiples of 20: 20, 40, 60, 80, …

The smallest number that appears in both lists is 60. So, the LCM of 15 and 20 is 60.

In both examples, the LCM represents the smallest common multiple of the given numbers, ensuring that it is divisible by both numbers without any remainder.

### How Do you Reduce Fractions to Lowest Terms?

Reducing fractions to their lowest terms, also known as simplifying fractions, involves dividing both the numerator (the top number) and the denominator (the bottom number) of the fraction by their greatest common divisor (GCD) or greatest common factor (GCF) to obtain an equivalent fraction in which the numerator and denominator have no common factors other than 1. Here are the steps for reducing fractions to their lowest terms, along with two examples:

Step 1: Find the GCD or GCF of the numerator and denominator. The GCD/GCF is the largest number that evenly divides both the numerator and the denominator.

Step 2: Divide both the numerator and denominator of the fraction by the GCD/GCF found in step 1.

Step 3: Write the simplified fraction.

Now, let’s look at two examples:

Example 1: Reduce 24/36 to its lowest terms:

Step 1: Find the GCD/GCF of 24 and 36.
The factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24.
The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, and 36.

The largest number that appears in both lists is 12. So, the GCD/GCF is 12.

Step 2: Divide both the numerator and denominator by 12.
(24 ÷ 12) / (36 ÷ 12) = 2/3

Step 3: Write the simplified fraction.
The simplified fraction is 2/3. So, 24/36 reduced to its lowest terms is 2/3.

Example 2: Reduce 15/25 to its lowest terms:

Step 1: Find the GCD/GCF of 15 and 25.
The factors of 15 are 1, 3, 5, and 15.
The factors of 25 are 1, 5, and 25.

The largest number that appears in both lists is 5. So, the GCD/GCF is 5.

Step 2: Divide both the numerator and denominator by 5.
(15 ÷ 5) / (25 ÷ 5) = 3/5

Step 3: Write the simplified fraction.
The simplified fraction is 3/5. So, 15/25 reduced to its lowest terms is 3/5.

In both examples, the fractions were reduced to their lowest terms by finding the GCD/GCF and dividing both the numerator and denominator by that common factor, resulting in equivalent fractions with the smallest possible numerator and denominator.

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