Find the LCM of Denominators: 8, 16, and 20

Q: How do I know which power of each prime to use?

Always take the highest power that appears in any factorization. Since 16 has $ 2^4 $, we use $ 2^4 $ (not $ 2^2 $ or $ 2^3 $).

Q: What if two numbers don't share any common factors?

If numbers are relatively prime (share no common factors), their LCM is simply their product. For example, LCM of 3 and 7 is $ 3 \times 7 = 21 $.

Q: How can I check if my LCM is correct?

Divide your LCM by each original number. If all divisions give whole numbers with no remainders, your LCM is correct! $ 80 ÷ 8 = 10 $, $ 80 ÷ 16 = 5 $, $ 80 ÷ 20 = 4 $ ✓

Prime Factorization with Multiple Numbers

What is the least common multiple of these denominators?

$\boxed{8}~~~\boxed{16} ~~~\boxed{20}$

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Step-by-step written solution

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Understand the problem

What is the least common multiple of these denominators?

$\boxed{8}~~~\boxed{16} ~~~\boxed{20}$

Step-by-step solution

To find the least common multiple (LCM) of $8$ , $16$ , and $20$ , find their prime factorizations:

$8 = 2^3$

$16 = 2^4$

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

The LCM is obtained by taking the highest power of each prime number:

$2^4$ from 16 and $5$ from 20.

The LCM is $2^4 \, \times \, 5 = 16 \, \times \, 5 = 80$ .

Final Answer

Key Points to Remember

Essential concepts to master this topic

Prime Factorization: Break each number into prime factors first
Technique: Take highest power of each prime: $2^4 \times 5 = 80$
Check: Verify 80 divides evenly by 8, 16, and 20 ✓

Common Mistakes

Avoid these frequent errors

Adding or multiplying all numbers together
Don't just multiply 8 × 16 × 20 = 2560! This gives a common multiple but not the LEAST common multiple. Always use prime factorization to find the smallest number that all denominators divide into evenly.

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 (2560), but not the least one! The LCM is the smallest number that all given numbers divide into evenly.

What if I don't remember how to find prime factors?

Start dividing by the smallest prime (2) repeatedly until you can't anymore, then try 3, then 5, etc. For example: $20 ÷ 2 = 10 ÷ 2 = 5$ , so $20 = 2^2 \times 5$ .

How do I know which power of each prime to use?

Always take the highest power that appears in any factorization. Since 16 has $2^4$ , we use $2^4$ (not $2^2$ or $2^3$ ).

What if two numbers don't share any common factors?

If numbers are relatively prime (share no common factors), their LCM is simply their product. For example, LCM of 3 and 7 is $3 \times 7 = 21$ .

How can I check if my LCM is correct?

Divide your LCM by each original number. If all divisions give whole numbers with no remainders, your LCM is correct! $80 ÷ 8 = 10$ , $80 ÷ 16 = 5$ , $80 ÷ 20 = 4$ ✓

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