Find the LCM of Three Numbers: 5, 10, and 20

Q: Why can't I just multiply all the numbers together?

Multiplying gives you a common multiple, but not the least one! For 5, 10, and 20, multiplying gives 1000, but the LCM is only 20. Prime factorization finds the smallest possible answer.

Q: How do I find prime factorization quickly?

Start with the smallest prime (2) and keep dividing: $ 20 ÷ 2 = 10 ÷ 2 = 5 $. Since 5 is prime, we get $ 20 = 2^2 \times 5 $.

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

LCM is the smallest number that all given numbers divide into. GCD is the largest number that divides into all given numbers. They're opposites!

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

No! The LCM must be at least as large as the biggest number you started with, since it needs to be divisible by that number.

Finding LCM with Prime Factorizations

What is the least common multiple of:

$\boxed{5}~~~\boxed{10} ~~~\boxed{20}$

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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:

$\boxed{5}~~~\boxed{10} ~~~\boxed{20}$

Step-by-step solution

To find the least common multiple (LCM) of $5$ , $10$ , and $20$ , find their prime factorizations:

$5 = 5$

$10 = 2 \, \times \, 5$

$20 = 2^2 \, \times \, 5$

$\times$ The LCM is determined by selecting the greatest power of each prime number:

$2^2$ from 20 and $5$ .

The LCM is $2^2 \, \times \, 5 = 4 \, \times \, 5 = 20$ .

Final Answer

Key Points to Remember

Essential concepts to master this topic

Prime Factorization: Break each number into its prime factors first
Technique: Take highest power of each prime: $2^2 \times 5 = 20$
Check: Verify 20 divides evenly by all three numbers: 20÷5=4, 20÷10=2, 20÷20=1 ✓

Common Mistakes

Avoid these frequent errors

Adding or multiplying all numbers together
Don't just multiply 5 × 10 × 20 = 1000! This gives you a common multiple but not the LEAST one. Always use prime factorization to find the smallest number that all original numbers divide into.

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 can't I just multiply all the numbers together?

Multiplying gives you a common multiple, but not the least one! For 5, 10, and 20, multiplying gives 1000, but the LCM is only 20. Prime factorization finds the smallest possible answer.

What if one number is already a multiple of the others?

Great observation! When one number contains all the prime factors of the others (like 20 contains factors of both 5 and 10), that largest number is your LCM.

How do I find prime factorization quickly?

Start with the smallest prime (2) and keep dividing: $20 ÷ 2 = 10 ÷ 2 = 5$ . Since 5 is prime, we get $20 = 2^2 \times 5$ .

What's the difference between LCM and GCD?

LCM is the smallest number that all given numbers divide into. GCD is the largest number that divides into all given numbers. They're opposites!

Can the LCM be smaller than the largest number?

No! The LCM must be at least as large as the biggest number you started with, since it needs to be divisible by that number.

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