Calculate LCM of Denominators: Finding the Least Common Multiple of 3, 5, and 7

Q: Why can I just multiply when the numbers are prime?

When numbers are prime (like 3, 5, and 7), they have no common factors except 1. So the LCM is simply their product! This shortcut only works with prime numbers.

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

You'd need to find the prime factorization of each number first, then use the highest power of each prime factor. But since 3, 5, and 7 are already prime, we can skip this step!

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

Divide your answer by each original number. If you get whole numbers every time, your LCM is correct! For example: 105 ÷ 3 = 35, 105 ÷ 5 = 21, 105 ÷ 7 = 15.

Q: Is there a smaller number that works?

No! The LCM is the smallest number that all denominators divide into evenly. Any smaller number won't work for all three.

Q: When do I need to find LCM of denominators?

You need LCM when adding or subtracting fractions with different denominators. The LCM becomes your common denominator!

Q: What if I got 70 or 35 as my answer?

Check your work! $ 70 = 2 \times 5 \times 7 $ (missing 3) and $ 35 = 5 \times 7 $ (missing 3). Neither divides evenly by 3, so they can't be the LCM.

LCM with Prime Number Factors

Given several denominators, what is their least common multiple?

$\boxed{5} \boxed{7} \boxed{3}$

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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?

$\boxed{5} \boxed{7} \boxed{3}$

Step-by-step solution

The least common multiple (LCM) of $5, 7, \text{ and } 3$ is the smallest positive integer that is divisible by each of these numbers.

First, calculate the LCM by multiplying the numbers, as they are all prime:

LCM is $5 \times 7 \times 3 = 105$ .

So the least common multiple is $105$ .

Final Answer

105

Key Points to Remember

Essential concepts to master this topic

Prime Numbers: When all denominators are prime, multiply them directly
Calculation: LCM of 3, 5, 7 equals $3 \times 5 \times 7 = 105$
Check: Verify 105 ÷ 3 = 35, 105 ÷ 5 = 21, 105 ÷ 7 = 15 ✓

Common Mistakes

Avoid these frequent errors

Adding the numbers instead of multiplying
Don't add 3 + 5 + 7 = 15! This gives a number that isn't divisible by all three denominators. Always multiply prime numbers together to find their LCM.

Practice Quiz

Test your knowledge with interactive questions

You have a pair of denominators, what is their least common multiple?

$\boxed 2~~~\boxed5$

FAQ

Everything you need to know about this question

Why can I just multiply when the numbers are prime?

When numbers are prime (like 3, 5, and 7), they have no common factors except 1. So the LCM is simply their product! This shortcut only works with prime numbers.

What if the numbers weren't all prime?

You'd need to find the prime factorization of each number first, then use the highest power of each prime factor. But since 3, 5, and 7 are already prime, we can skip this step!

How do I check if my LCM is correct?

Divide your answer by each original number. If you get whole numbers every time, your LCM is correct! For example: 105 ÷ 3 = 35, 105 ÷ 5 = 21, 105 ÷ 7 = 15.

Is there a smaller number that works?

No! The LCM is the smallest number that all denominators divide into evenly. Any smaller number won't work for all three.

When do I need to find LCM of denominators?

You need LCM when adding or subtracting fractions with different denominators. The LCM becomes your common denominator!

What if I got 70 or 35 as my answer?

Check your work! $70 = 2 \times 5 \times 7$ (missing 3) and $35 = 5 \times 7$ (missing 3). Neither divides evenly by 3, so they can't be the LCM.

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