LCM Problem: Finding the Least Common Multiple of 5 and 9

Q: Why isn't the LCM just 5×9=45?

Great observation! For coprime numbers (numbers that share no common factors except 1), the LCM equals their product. Since 5 and 9 share no common factors, $ 5 \times 9 = 45 $ is correct!

Q: What if I can't list all the multiples?

Try the prime factorization method! Find $ 5 = 5^1 $ and $ 9 = 3^2 $, then multiply the highest powers: $ 5^1 \times 3^2 = 5 \times 9 = 45 $.

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

Stop when you find the first number that appears in both lists. That's your LCM! In this case, 45 appears in both the multiples of 5 and multiples of 9.

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 5 and 9: LCM = 45, GCD = 1.

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

Never! The LCM must be at least as large as the bigger number, since it needs to be divisible by both original numbers. The LCM of 5 and 9 must be ≥ 9.

Finding LCM with Coprime Numbers

Find the least common multiple of 5 and 9.

$\boxed 5~~~\boxed 9$

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Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Find the least common multiple of 5 and 9.

$\boxed 5~~~\boxed 9$

Step-by-step solution

To find the least common multiple (LCM) of 5 and 9, list the multiples of each number and find the smallest multiple they have in common.

Multiples of 5: $5, 10, 15, 20, 25, 30, 35, 40, 45, \ldots$

Multiples of 9: $9, 18, 27, 36, 45, \ldots$

The smallest common multiple is $45$ .

Final Answer

Key Points to Remember

Essential concepts to master this topic

Definition: LCM is the smallest positive number divisible by both numbers
Method: List multiples of each: 5→5,10,15,20,25,30,35,40,45... and 9→9,18,27,36,45...
Verification: Check that 45÷5=9 and 45÷9=5 with no remainder ✓

Common Mistakes

Avoid these frequent errors

Confusing LCM with multiplication
Don't just multiply 5×9=45 without checking! While this works for coprime numbers (like 5 and 9), it fails for numbers with common factors. Always verify by listing multiples or using prime factorization to confirm your answer is truly the least common multiple.

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 just 5×9=45?

Great observation! For coprime numbers (numbers that share no common factors except 1), the LCM equals their product. Since 5 and 9 share no common factors, $5 \times 9 = 45$ is correct!

What if I can't list all the multiples?

Try the prime factorization method! Find $5 = 5^1$ and $9 = 3^2$ , then multiply the highest powers: $5^1 \times 3^2 = 5 \times 9 = 45$ .

How do I know when to stop listing multiples?

Stop when you find the first number that appears in both lists. That's your LCM! In this case, 45 appears in both the multiples of 5 and multiples of 9.

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 5 and 9: LCM = 45, GCD = 1.

Can the LCM ever be smaller than both numbers?

Never! The LCM must be at least as large as the bigger number, since it needs to be divisible by both original numbers. The LCM of 5 and 9 must be ≥ 9.

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