The least common multiple (LCM) of two numbers is the smallest positive integer that is divisible by both numbers. To find the LCM of 75 and 8, we first need to understand what a least common multiple is and how to calculate it for two given numbers.

## What is a Least Common Multiple?

The least common multiple of two integers is the smallest positive number that is divisible by both integers. Some key properties of LCMs:

- The LCM is always a multiple of the numbers involved.
- The LCM is the smallest (least) common multiple of the numbers.
- To find the LCM, find all the common multiples and select the smallest one.

For example, the LCM of 4 and 6 is 12. This is because:

- 12 is a common multiple of both 4 and 6.
- 12 is the smallest number that satisfies this.

Some other examples:

- LCM(2, 3) = 6
- LCM(5, 7) = 35
- LCM(12, 18) = 36

So to find the LCM of any two numbers, we need to find their common multiples and then locate the smallest common multiple.

## How to Find the LCM of Two Numbers

There are a couple of different ways to find the LCM of two numbers:

### List Multiples Method

To find LCM(a, b):

- List all multiples of a
- List all multiples of b
- Identify the smallest common multiple on both lists. This is the LCM(a, b).

For example, to find LCM(15, 20) using this method:

- Multiples of 15 are: 15, 30, 45, 60, 75…
- Multiples of 20 are: 20, 40, 60, 80, 100…

The smallest common multiple is 60. Therefore, LCM(15, 20) = 60.

### Prime Factorization Method

To find LCM(a, b) using prime factorization:

- Express a and b in terms of their prime factors
- Take the highest power of each prime number
- Multiply all the prime factors together. This is the LCM(a, b).

For example, to find LCM(24, 36):

- Prime factors of 24 are 2
^{3}x 3 - Prime factors of 36 are 2
^{2}x 3^{2}

Taking the highest power of each prime factor:

- 2
^{3} - 3
^{2}

Multiplying these together:

2^{3} x 3^{2} = 72

Therefore, LCM(24, 36) = 72

This tends to be a quicker method compared to listing all multiples.

## Finding the LCM of 75 and 8

Now let’s use these methods to find the LCM of 75 and 8:

### List Multiples Method

- Multiples of 75 are: 75, 150, 225, 300…
- Multiples of 8 are: 8, 16, 24, 32…

The smallest common multiple is 300.

Therefore, LCM(75, 8) = 300.

### Prime Factorization Method

The prime factors are:

- 75 = 3 x 5
^{2} - 8 = 2
^{3}

Taking the highest powers:

- 3
- 5
^{2} - 2
^{3}

Multiplying these together:

3 x 5^{2} x 2^{3} = 300

Therefore, LCM(75, 8) = 300

## Conclusion

Using both the list multiples method and prime factorization, we determine that the LCM of 75 and 8 is 300.

The key steps to finding the LCM of two numbers are:

- List the multiples of each number
- Identify the smallest common multiple
- Or, express the numbers in terms of prime factors
- Take the highest power of each prime factor
- Multiply the prime factors together

Knowing how to find the LCM is useful for many math applications like solving proportion problems or finding the least common denominator. The LCM represents the smallest common size or space that both original numbers fit into. Being able to determine LCMs is an important math skill.