Find the LCM of Denominators: 14, 15, and 3 - Step by Step

Q: Why isn't the answer just 14 × 15 × 3 = 630?

That's a common multiple but not the least common multiple! The LCM is the smallest number that all denominators divide into evenly. Since 210 works and is smaller than 630, it's the correct LCM.

Q: How do I find prime factors quickly?

Start with the smallest prime (2) and keep dividing: 14 = 2 × 7, 15 = 3 × 5, 3 = 3. Stop when you can't divide anymore!

Q: What if two numbers share the same prime factor?

Use the highest power of that prime! For example, if you had 12 and 18, both have factor 3, but you only need $ 3^2 $ (the higher power) in your LCM.

Q: How can I check if 210 is really the LCM?

Divide 210 by each original number: 210 ÷ 14 = 15 ✓, 210 ÷ 15 = 14 ✓, 210 ÷ 3 = 70 ✓. All give whole numbers, so 210 is correct!

Prime Factorization with Multiple Denominators

Given several denominators, what is their least common multiple?

$\boxed{14} \boxed{15} \boxed{3}$

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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?

$\boxed{14} \boxed{15} \boxed{3}$

Step-by-step solution

The least common multiple (LCM) of $14, 15, \text{ and } 3$ is the smallest positive integer that is divisible by each of these numbers.

Using the prime factors, we find:

The prime factors of $14$ are 2 and 7.
The prime factors of $15$ are 3 and 5.
The prime factors of $3$ is 3.

The LCM will be $2 \times 3 \times 5 \times 7 = 210$ .

Therefore, the least common multiple is $105$ .

Final Answer

105

Key Points to Remember

Essential concepts to master this topic

Prime Factorization: Break each number into prime factors first
LCM Method: Use highest power of each prime: $2^1 \times 3^1 \times 5^1 \times 7^1 = 210$
Verification: Check that 210 ÷ 14 = 15, 210 ÷ 15 = 14, 210 ÷ 3 = 70 ✓

Common Mistakes

Avoid these frequent errors

Multiplying all numbers together instead of finding LCM
Don't just multiply 14 × 15 × 3 = 630! This gives a common multiple but not the LEAST common multiple. Always find prime factors and use the highest power of each prime to get the smallest possible answer.

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 isn't the answer just 14 × 15 × 3 = 630?

That's a common multiple but not the least common multiple! The LCM is the smallest number that all denominators divide into evenly. Since 210 works and is smaller than 630, it's the correct LCM.

How do I find prime factors quickly?

Start with the smallest prime (2) and keep dividing: 14 = 2 × 7, 15 = 3 × 5, 3 = 3. Stop when you can't divide anymore!

What if two numbers share the same prime factor?

Use the highest power of that prime! For example, if you had 12 and 18, both have factor 3, but you only need $3^2$ (the higher power) in your LCM.

Why does the explanation say 105 but the work shows 210?

There's an error in the explanation! The correct calculation $2 \times 3 \times 5 \times 7 = 210$ gives 210, but the final statement incorrectly says 105. Always trust your calculation over the written conclusion.

How can I check if 210 is really the LCM?

Divide 210 by each original number: 210 ÷ 14 = 15 ✓, 210 ÷ 15 = 14 ✓, 210 ÷ 3 = 70 ✓. All give whole numbers, so 210 is correct!

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