Find the LCM of Denominators: Calculating LCM(2, 3, 7)

Q: Why can't I just add the denominators together?

Adding gives you 2 + 3 + 7 = 12, but 12 isn't divisible by 7! The LCM must be divisible by every denominator, not just some of them.

Q: What if the numbers weren't all prime?

You'd need to find the prime factorization of each number, then take the highest power of each prime factor. For primes like 2, 3, and 7, just multiply them together!

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

LCM (Least Common Multiple) finds the smallest number that contains all factors. GCD (Greatest Common Divisor) finds the largest number that divides into all given numbers.

Prime Factor LCM with Multiple Denominators

Given three denominators, what is their least common multiple?

$\boxed{3}~~~\boxed{7}~~~\boxed{2}$

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Step-by-step written solution

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Understand the problem

Given three denominators, what is their least common multiple?

$\boxed{3}~~~\boxed{7}~~~\boxed{2}$

Step-by-step solution

To find the least common multiple (LCM) of the denominators 3, 7, and 2, we find the smallest positive integer that is divisible by all three numbers. The prime factors are:

$3 = 3$

$7 = 7$

$2 = 2$

Since all numbers are primes, the least common multiple is simply their product:

$3 \times 7 \times 2 = 42$

Final Answer

Key Points to Remember

Essential concepts to master this topic

Definition: LCM is smallest number divisible by all given denominators
Prime Method: Since 2, 3, 7 are all prime, multiply directly: $2 \times 3 \times 7 = 42$
Verification: Check that 42 ÷ 2 = 21, 42 ÷ 3 = 14, 42 ÷ 7 = 6 ✓

Common Mistakes

Avoid these frequent errors

Adding denominators instead of finding LCM
Don't add 2 + 3 + 7 = 12! This gives a number that isn't divisible by all denominators (12 ÷ 7 doesn't work). Always 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 add the denominators together?

Adding gives you 2 + 3 + 7 = 12, but 12 isn't divisible by 7! The LCM must be divisible by every denominator, not just some of them.

What if the numbers weren't all prime?

You'd need to find the prime factorization of each number, then take the highest power of each prime factor. For primes like 2, 3, and 7, just multiply them together!

How do I know 42 is really the smallest?

Check smaller numbers: 21 works for 3 and 7 but not 2. 14 works for 2 and 7 but not 3. Only 42 works for all three denominators.

Is there a faster way to check my answer?

Yes! Divide your LCM by each original denominator. If you get whole numbers every time (like 42÷2=21, 42÷3=14, 42÷7=6), your answer is correct.

What's the difference between LCM and GCD?

LCM (Least Common Multiple) finds the smallest number that contains all factors. GCD (Greatest Common Divisor) finds the largest number that divides into all given numbers.

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