Find the LCM: Calculating Least Common Multiple of 6 and 9

Q: Can I use prime factorization to find LCM?

Yes! Break down each number: $ 6 = 2 \times 3 $ and $ 9 = 3^2 $. The LCM uses the highest power of each prime: $ 2^1 \times 3^2 = 18 $.

Q: What if one number is a multiple of the other?

When one number divides evenly into another, the larger number is always the LCM. For example, LCM of 4 and 12 is just 12 since 12 ÷ 4 = 3.

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

Stop as soon as you find the first number that appears in both lists. That's your LCM! You don't need to continue beyond the first match.

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

Yes! Use the formula: $ \text{LCM} = \frac{a \times b}{\text{GCF}(a,b)} $. For 6 and 9: $ \text{LCM} = \frac{6 \times 9}{3} = 18 $.

LCM with Prime Factorization

What is the least common multiple of these two numbers?

$\boxed{6}~~~\boxed{9}$

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Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

What is the least common multiple of these two numbers?

$\boxed{6}~~~\boxed{9}$

Step-by-step solution

To find the least common multiple (LCM) of $6$ and $9$ , we list the multiples of each number:

Multiples of $6$ are $6, 12, 18, 24, \ldots$
Multiples of $9$ are $9, 18, 27, \ldots$

The smallest common multiple is $18$ .

Final Answer

Key Points to Remember

Essential concepts to master this topic

Definition: LCM is the smallest positive number divisible by both numbers
Technique: List multiples of each: 6: 6, 12, 18... and 9: 9, 18, 27...
Check: Verify 18 ÷ 6 = 3 and 18 ÷ 9 = 2 with no remainder ✓

Common Mistakes

Avoid these frequent errors

Confusing LCM with GCF
Don't find the greatest common factor (GCF = 3) when asked for LCM! GCF finds the largest number that divides both, while LCM finds the smallest number that both divide into. Always look for the smallest common multiple, not the largest common divisor.

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 isn't the LCM of 6 and 9 equal to 6 × 9 = 54?

Multiplying the numbers gives you a common multiple, but not necessarily the least one! Since 6 and 9 share a common factor of 3, their LCM is smaller: $\frac{6 \times 9}{3} = 18$ .

Can I use prime factorization to find LCM?

Yes! Break down each number: $6 = 2 \times 3$ and $9 = 3^2$ . The LCM uses the highest power of each prime: $2^1 \times 3^2 = 18$ .

What if one number is a multiple of the other?

When one number divides evenly into another, the larger number is always the LCM. For example, LCM of 4 and 12 is just 12 since 12 ÷ 4 = 3.

How do I know when to stop listing multiples?

Stop as soon as you find the first number that appears in both lists. That's your LCM! You don't need to continue beyond the first match.

Is there a faster way than listing all multiples?

Yes! Use the formula: $\text{LCM} = \frac{a \times b}{\text{GCF}(a,b)}$ . For 6 and 9: $\text{LCM} = \frac{6 \times 9}{3} = 18$ .

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