Find the LCM of Denominators: 6, 8, and 10 - Step by Step

Q: Why can't I just multiply 6 × 8 × 10 to get the LCM?

While $6 \times 8 \times 10 = 480$ is a common multiple, it's not the least common multiple! The LCM is the smallest number that all denominators divide into evenly.

Q: What if I forget to use the highest power of a prime?

You'll get a number that's too small! For example, using $2^1$ instead of $2^3$ would give you 30, but 8 doesn't divide evenly into 30. Always check each prime's highest power.

Q: How do I know which power of each prime to use?

Look at each prime factor across all numbers and pick the highest power. Since 8 = $2^3$, we need $2^3$ in our LCM, not just $2^1$ from 6 and 10.

Q: Can I use a different method to find the LCM?

Yes! You can also use the listing method (write multiples of each number until you find a common one) or division method. But prime factorization is usually fastest for larger numbers.

Q: What if two numbers share no common factors?

If numbers are relatively prime (share no common factors except 1), their LCM is simply their product. For example, LCM of 7 and 11 is $7 \times 11 = 77$.

Prime Factorization with Multiple Numbers

Given several denominators, what is their least common multiple?

$\boxed6~~~\boxed8 ~~~\boxed{10 }$

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Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Given several denominators, what is their least common multiple?

$\boxed6~~~\boxed8 ~~~\boxed{10 }$

Step-by-step solution

To find the least common multiple (LCM) of the denominators $6, 8, 10$ , we need to consider each prime factor of these numbers at their highest power:

$6 = 2 \times 3$
$8 = 2^3$
$10 = 2 \times 5$

Therefore, the LCM is:

$2^3 \times 3 \times 5 = 120$

So, the least common multiple of $6, 8, 10$ is $120$ .

Final Answer

120

Key Points to Remember

Essential concepts to master this topic

Prime Factorization: Break each number into prime factors first
Technique: Use highest power of each prime: $2^3 \times 3 \times 5 = 120$
Check: Verify 120 divides evenly by 6, 8, and 10 ✓

Common Mistakes

Avoid these frequent errors

Multiplying all numbers together instead of using prime factorization
Don't multiply 6 × 8 × 10 = 480! This gives you a common multiple but not the LEAST common multiple. Always find prime factors and use the highest power of each prime to get the smallest possible answer.

Practice Quiz

Test your knowledge with interactive questions

You have a pair of denominators, what is their least common multiple?

$\boxed 2~~~\boxed5$

FAQ

Everything you need to know about this question

Why can't I just multiply 6 × 8 × 10 to get the LCM?

While $6 \times 8 \times 10 = 480$ is a common multiple, it's not the least common multiple! The LCM is the smallest number that all denominators divide into evenly.

What if I forget to use the highest power of a prime?

You'll get a number that's too small! For example, using $2^1$ instead of $2^3$ would give you 30, but 8 doesn't divide evenly into 30. Always check each prime's highest power.

How do I know which power of each prime to use?

Look at each prime factor across all numbers and pick the highest power. Since 8 = $2^3$ , we need $2^3$ in our LCM, not just $2^1$ from 6 and 10.

Can I use a different method to find the LCM?

Yes! You can also use the listing method (write multiples of each number until you find a common one) or division method. But prime factorization is usually fastest for larger numbers.

What if two numbers share no common factors?

If numbers are relatively prime (share no common factors except 1), their LCM is simply their product. For example, LCM of 7 and 11 is $7 \times 11 = 77$ .

Why is finding the LCM important for fractions?

When adding or subtracting fractions with different denominators, you need a common denominator. The LCM gives you the smallest possible common denominator, making calculations easier!

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