Find the LCM of Three Numbers: 8, 14, and 20

Q: What if I forget to use the highest power of a prime?

You'll get a number that's too small! For example, using $ 2^2 $ instead of $ 2^3 $ would give 140, but 140 ÷ 8 doesn't equal a whole number.

Q: How do I find prime factorization quickly?

Start with the smallest prime (2) and keep dividing: 8 = 2×4 = 2×2×2 = 2³. Then try 3, 5, 7, etc. until you can't divide anymore.

LCM Calculation with Prime Factorization

Find the least common multiple (LCM) of these numbers:

$\boxed{8} ~~~ \boxed{14} ~~~ \boxed{20}$

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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 (LCM) of these numbers:

$\boxed{8} ~~~ \boxed{14} ~~~ \boxed{20}$

Step-by-step solution

To find the least common multiple (LCM) of 8, 14, and 20, determine their prime factorization:

Prime factors of 8: $8 = 2^3$

Prime factors of 14: $14 = 2^1 \times 7^1$

Prime factors of 20: $20 = 2^2 \times 5^1$

The LCM is the product of the highest powers of all prime factors:

$2^3 \times 7^1 \times 5^1 = 8 \times 35 = 280$

Final Answer

280

Key Points to Remember

Essential concepts to master this topic

Prime Factorization: Break each number into prime factors with powers
Technique: Take highest power of each prime: $2^3 \times 7^1 \times 5^1 = 280$
Check: LCM divides evenly: 280÷8=35, 280÷14=20, 280÷20=14 ✓

Common Mistakes

Avoid these frequent errors

Multiplying all the numbers together
Don't just multiply 8 × 14 × 20 = 2240! This gives you a common multiple, but not the LEAST common multiple. Always find prime factorizations and use the highest power of each prime factor.

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 8 × 14 × 20 = 2240 gives you a common multiple, but not the least one! The LCM is always smaller because it uses only the highest powers needed.

What if I forget to use the highest power of a prime?

You'll get a number that's too small! For example, using $2^2$ instead of $2^3$ would give 140, but 140 ÷ 8 doesn't equal a whole number.

How do I find prime factorization quickly?

Start with the smallest prime (2) and keep dividing: 8 = 2×4 = 2×2×2 = 2³. Then try 3, 5, 7, etc. until you can't divide anymore.

Can I use a different method to find LCM?

Yes! You can list multiples of each number and find the smallest common one, but prime factorization is faster for larger numbers like these.

How do I know my LCM is correct?

Your LCM should divide evenly into each original number. Check: 280 ÷ 8 = 35, 280 ÷ 14 = 20, and 280 ÷ 20 = 14 - all whole numbers!

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