Find the LCM of Denominators: Calculating LCM(3, 4, 6)

Q: Why isn't 12 the correct answer if it's a common multiple?

You're right that 12 is a common multiple of 3, 4, and 6! However, we need the least common multiple. Since $ 12 \div 4 = 3 $ but 4 doesn't divide 12 evenly (12÷4=3 with no remainder), 12 works for 3 and 6 but not 4.

Q: How do I know when to stop listing multiples?

Stop when you find the first number that appears in all lists. Once you see 24 in the multiples of 3, 4, and 6, that's your LCM! No need to continue further.

Q: Is there a faster way than listing all multiples?

Yes! You can use prime factorization: $ 3 = 3 $, $ 4 = 2^2 $, $ 6 = 2 \times 3 $. Take the highest power of each prime: $ 2^2 \times 3 = 12 \times 2 = 24 $.

Q: Why do we need the LCM for fractions?

The LCM of denominators becomes your common denominator when adding or subtracting fractions! For example, $ \frac{1}{3} + \frac{1}{4} + \frac{1}{6} $ needs denominator 24 to solve.

LCM Calculation with Multiple Numbers

Given several denominators, what is their least common multiple?

$\boxed{3} \boxed{4} \boxed{6}$

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Step-by-step written solution

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Understand the problem

Given several denominators, what is their least common multiple?

$\boxed{3} \boxed{4} \boxed{6}$

Step-by-step solution

The least common multiple (LCM) of $3, 4, \text{ and } 6$ is the smallest positive integer that is divisible by each of these numbers.

First, list the multiples of each number:

Multiples of $3$ : 3, 6, 9, 12, 15, 18, 21, 24, ...
Multiples of $4$ : 4, 8, 12, 16, 20, 24, ...
Multiples of $6$ : 6, 12, 18, 24, ...

The common multiples of $3, 4, \text{ and } 6$ are 12, 24, ...

The smallest common multiple is $24$ .

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 of each: 3→24, 4→24, 6→24 first match
Check: Verify 24÷3=8, 24÷4=6, 24÷6=4 all divide evenly ✓

Common Mistakes

Avoid these frequent errors

Confusing LCM with GCF or choosing largest given number
Don't pick 6 as the answer just because it's the largest denominator = wrong! The LCM must be divisible by ALL numbers, not just be the biggest. Always find the smallest number that ALL given numbers divide into evenly.

Practice Quiz

Test your knowledge with interactive questions

Without calculating, determine whether the quotient in the division exercise is less than 1 or not:

$$ 5:6= $$

FAQ

Everything you need to know about this question

Why isn't 12 the correct answer if it's a common multiple?

You're right that 12 is a common multiple of 3, 4, and 6! However, we need the least common multiple. Since $12 \div 4 = 3$ but 4 doesn't divide 12 evenly (12÷4=3 with no remainder), 12 works for 3 and 6 but not 4.

How do I know when to stop listing multiples?

Stop when you find the first number that appears in all lists. Once you see 24 in the multiples of 3, 4, and 6, that's your LCM! No need to continue further.

Is there a faster way than listing all multiples?

Yes! You can use prime factorization: $3 = 3$ , $4 = 2^2$ , $6 = 2 \times 3$ . Take the highest power of each prime: $2^2 \times 3 = 12 \times 2 = 24$ .

What if one number is a multiple of another?

Great observation! When one number divides evenly into another (like 3 divides into 6), the larger number might be the LCM. But always check all numbers - here 6 doesn't work because $6 \div 4$ doesn't give a whole number.

Why do we need the LCM for fractions?

The LCM of denominators becomes your common denominator when adding or subtracting fractions! For example, $\frac{1}{3} + \frac{1}{4} + \frac{1}{6}$ needs denominator 24 to solve.

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