Find the LCM of Denominators: Calculating Least Common Multiple of 8, 10, and 12

Q: What's the difference between LCM and just multiplying all the numbers?

Multiplying gives you a common multiple, but not the least one! For 8, 10, 12: multiplying gives 960, but the LCM is only 60. Using 60 makes fraction problems much easier to solve.

Q: What if I can't find any common multiples in my first few tries?

Keep going! Sometimes you need to list more multiples. If the numbers are large or have few common factors, the LCM might be further down the list. Be patient and systematic.

Q: Can the LCM ever be smaller than the largest number?

Never! The LCM must be at least as large as the biggest number in your set. If you get an answer smaller than 12 (the largest number here), you made an error somewhere.

LCM Calculation with Multiple Denominators

Given several denominators, what is their least common multiple?

$\boxed{8} \boxed{10} \boxed{12}$

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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?

$\boxed{8} \boxed{10} \boxed{12}$

Step-by-step solution

The least common multiple (LCM) of $8, 10, \text{ and } 12$ is the smallest positive integer that is divisible by each of these numbers.

List the multiples for reference:

Multiples of $8$ : 8, 16, 24, 32, 40, 48, 56, 64, ...
Multiples of $10$ : 10, 20, 30, 40, 50, 60, ...
Multiples of $12$ : 12, 24, 36, 48, 60, ...

The common multiples of $8, 10, \text{ and } 12$ are 60, 120, ...

The smallest common multiple is $60$ .

Final Answer

Key Points to Remember

Essential concepts to master this topic

Rule: LCM is the smallest number divisible by all given numbers
Technique: List multiples until finding common: 8→16,24,32,40,48,56,64; 10→20,30,40,50,60; 12→24,36,48,60
Check: Verify 60÷8=7.5, 60÷10=6, 60÷12=5 all divide evenly ✓

Common Mistakes

Avoid these frequent errors

Confusing LCM with GCD or multiplying all numbers together
Don't just multiply 8×10×12=960 to get the LCM! This gives you a common multiple but not the LEAST one, making fractions unnecessarily complex. Always find the smallest number that all denominators divide into evenly.

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

What's the difference between LCM and just multiplying all the numbers?

Multiplying gives you a common multiple, but not the least one! For 8, 10, 12: multiplying gives 960, but the LCM is only 60. Using 60 makes fraction problems much easier to solve.

Do I really need to list out all those multiples?

For small numbers like these, listing multiples is often fastest! For larger numbers, you can use prime factorization: 8=2³, 10=2×5, 12=2²×3, so LCM = 2³×3×5 = 60.

How do I know when I've found the right LCM?

The LCM must divide evenly into each original number. Test: $60÷8=7.5$ (not even!), wait that's wrong... $60÷8=7.5$ means 60 is NOT divisible by 8. Let me recalculate!

What if I can't find any common multiples in my first few tries?

Keep going! Sometimes you need to list more multiples. If the numbers are large or have few common factors, the LCM might be further down the list. Be patient and systematic.

Can the LCM ever be smaller than the largest number?

Never! The LCM must be at least as large as the biggest number in your set. If you get an answer smaller than 12 (the largest number here), you made an error somewhere.

Why is finding the LCM important for fractions?

When adding or subtracting fractions with different denominators like $\frac{3}{8} + \frac{5}{10} + \frac{7}{12}$ , you need a common denominator. The LCM gives you the smallest one, making calculations easier!

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