Find the LCM of Denominators: 2, 5, and 7

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

The LCM must be divisible by each original number. Adding gives 14, but 14 ÷ 5 = 2.8 (not a whole number). Only multiplication ensures the result contains all needed factors.

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

For non-prime numbers, you'd need to find prime factorizations first, then take the highest power of each prime factor. But since 2, 5, and 7 are already prime, we simply multiply them!

Q: Is there a faster way to find LCM?

When all numbers are different primes (like 2, 5, 7), the fastest method is direct multiplication. For numbers with common factors, you'd use prime factorization instead.

Q: Why is this useful for fractions?

Finding the LCM of denominators helps you add or subtract fractions! You convert all fractions to have denominator 70, making calculations much easier.

LCM Calculation with Prime Numbers

Given several denominators, what is their least common multiple?

$\boxed{2}~~~\boxed{5} ~~~\boxed{7}$

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Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Given several denominators, what is their least common multiple?

$\boxed{2}~~~\boxed{5} ~~~\boxed{7}$

Step-by-step solution

To find the least common multiple (LCM) of $2$ , $5$ , and $7$ , start with the prime factorizations:

$2$ , $5$ , and $7$ , as they all are primes.

The LCM is simply their product: $2 \, \times \, 5 \, \times \, 7 = 70$ .

Final Answer

Key Points to Remember

Essential concepts to master this topic

Prime Rule: When numbers are all prime, LCM equals their product
Technique: Multiply directly: $2 \times 5 \times 7 = 70$
Check: Verify 70 divides by 2, 5, and 7 with no remainders ✓

Common Mistakes

Avoid these frequent errors

Adding instead of multiplying prime denominators
Don't add primes like 2 + 5 + 7 = 14! This gives a number that doesn't contain all prime factors needed. Always multiply all prime numbers together to get the true LCM.

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 numbers together?

The LCM must be divisible by each original number. Adding gives 14, but 14 ÷ 5 = 2.8 (not a whole number). Only multiplication ensures the result contains all needed factors.

What if the numbers weren't all prime?

For non-prime numbers, you'd need to find prime factorizations first, then take the highest power of each prime factor. But since 2, 5, and 7 are already prime, we simply multiply them!

How do I know 70 is really the smallest common multiple?

Check smaller multiples: 35 doesn't divide by 2, and 14 doesn't divide by 5. Since our numbers are all different primes, their LCM must be their product - no smaller number works!

Is there a faster way to find LCM?

When all numbers are different primes (like 2, 5, 7), the fastest method is direct multiplication. For numbers with common factors, you'd use prime factorization instead.

Why is this useful for fractions?

Finding the LCM of denominators helps you add or subtract fractions! You convert all fractions to have denominator 70, making calculations much easier.

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