Find the LCM of Numerators: Calculating LCM(3, 5, 7)

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

Adding gives you a sum, not a multiple! The LCM must be divisible by each number. Since 15 ÷ 7 doesn't work, 15 isn't the LCM.

Q: What makes this problem easier than other LCM questions?

Since 3, 5, and 7 are all prime numbers, they share no common factors except 1. This means their LCM is simply their product: $ 3 \times 5 \times 7 = 105 $.

Q: How do I know 105 is really the smallest common multiple?

Check that 105 is divisible by each number: 105 ÷ 3 = 35 ✓105 ÷ 5 = 21 ✓105 ÷ 7 = 15 ✓Any smaller number won't divide evenly by all three!

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

Then you'd need to find the highest power of each prime factor. But here, each number appears only once as $ 3^1 $, $ 5^1 $, and $ 7^1 $.

Q: Is there a faster way than prime factorization?

For prime numbers like these, just multiply them directly! Prime factorization shows why this works, but $ 3 \times 5 \times 7 = 105 $ is the quickest method.

LCM Calculation with Prime Numbers

Determine the least common multiple (LCM) of the following numerators:

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

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Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Determine the least common multiple (LCM) of the following numerators:

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

Step-by-step solution

To find the least common multiple (LCM) of the numbers 3, 7, and 5, we use the prime factorization method:

Prime factors of each number:

3: $3^1$
7: $7^1$
5: $5^1$

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

$3^1 \times 7^1 \times 5^1 = 3 \times 7 \times 5 = 105$

Final Answer

105

Key Points to Remember

Essential concepts to master this topic

Prime Factorization: When all numbers are prime, LCM equals their product
Technique: $3^1 \times 5^1 \times 7^1 = 3 \times 5 \times 7 = 105$
Check: 105 ÷ 3 = 35, 105 ÷ 5 = 21, 105 ÷ 7 = 15 ✓

Common Mistakes

Avoid these frequent errors

Adding prime numbers instead of multiplying
Don't add 3 + 5 + 7 = 15! This gives a number that's not divisible by all three. When finding LCM of prime numbers, they share no common factors, so you must multiply them together: 3 × 5 × 7 = 105.

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't I just add the prime numbers together?

Adding gives you a sum, not a multiple! The LCM must be divisible by each number. Since 15 ÷ 7 doesn't work, 15 isn't the LCM.

What makes this problem easier than other LCM questions?

Since 3, 5, and 7 are all prime numbers, they share no common factors except 1. This means their LCM is simply their product: $3 \times 5 \times 7 = 105$ .

How do I know 105 is really the smallest common multiple?

Check that 105 is divisible by each number:

105 ÷ 3 = 35 ✓
105 ÷ 5 = 21 ✓
105 ÷ 7 = 15 ✓

Any smaller number won't divide evenly by all three!

What if the numbers weren't all prime?

Then you'd need to find the highest power of each prime factor. But here, each number appears only once as $3^1$ , $5^1$ , and $7^1$ .

Is there a faster way than prime factorization?

For prime numbers like these, just multiply them directly! Prime factorization shows why this works, but $3 \times 5 \times 7 = 105$ is the quickest method.

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