Find the LCM: Calculating Least Common Multiple of 3 and 9

Q: Why isn't the LCM of 3 and 9 equal to 3 × 9 = 27?

The formula LCM = (a × b) ÷ GCD applies here! Since 9 is already a multiple of 3, their GCD is 3, so LCM = (3 × 9) ÷ 3 = 9, not 27.

Q: How do I know when one number is a multiple of another?

Divide the larger number by the smaller one. If you get a whole number with no remainder, then the larger number is a multiple of the smaller one.

Q: What's the difference between LCM and GCD?

LCM (Least Common Multiple) is the smallest number both divide into, while GCD (Greatest Common Divisor) is the largest number that divides both. For 3 and 9: LCM = 9, GCD = 3.

Q: Can the LCM ever be smaller than both numbers?

Never! The LCM must be divisible by both numbers, so it's always greater than or equal to the larger of the two numbers.

LCM Calculation with Divisor Relationships

What is the least common multiple of these two numbers?

$\boxed{3}~~~\boxed{9}$

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Step-by-step written solution

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Understand the problem

What is the least common multiple of these two numbers?

$\boxed{3}~~~\boxed{9}$

Step-by-step solution

To find the least common multiple (LCM) of $3$ and $9$ , we list the multiples of each number:

Multiples of $3$ are $3, 6, 9, 12, 15, \ldots$
Multiples of $9$ are $9, 18, 27, \ldots$

The smallest common multiple is $9$ .

Final Answer

Key Points to Remember

Essential concepts to master this topic

Definition: LCM is the smallest number divisible by both given numbers
Method: List multiples of each: 3, 6, 9, 12... and 9, 18, 27...
Check: Verify 9 ÷ 3 = 3 and 9 ÷ 9 = 1 both exactly ✓

Common Mistakes

Avoid these frequent errors

Confusing LCM with GCD or choosing the larger number without checking
Don't just pick 9 because it's bigger = wrong reasoning! The larger number isn't always the LCM. Always check if the larger number is actually a multiple of the smaller number first.

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 the LCM of 3 and 9 equal to 3 × 9 = 27?

The formula LCM = (a × b) ÷ GCD applies here! Since 9 is already a multiple of 3, their GCD is 3, so LCM = (3 × 9) ÷ 3 = 9, not 27.

How do I know when one number is a multiple of another?

Divide the larger number by the smaller one. If you get a whole number with no remainder, then the larger number is a multiple of the smaller one.

Is there a faster way than listing all multiples?

Yes! When one number divides evenly into another (like 3 divides into 9), the LCM is simply the larger number. Check: $9 ÷ 3 = 3$ exactly.

What's the difference between LCM and GCD?

LCM (Least Common Multiple) is the smallest number both divide into, while GCD (Greatest Common Divisor) is the largest number that divides both. For 3 and 9: LCM = 9, GCD = 3.

Can the LCM ever be smaller than both numbers?

Never! The LCM must be divisible by both numbers, so it's always greater than or equal to the larger of the two numbers.

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