Find the LCM of Denominators: Calculating LCM(3,4)

Q: Why can't I just multiply 3 × 4 = 12?

You can when the numbers are coprime (share no common factors), like 3 and 4! But this shortcut doesn't always work. For numbers like 6 and 8, multiplying gives 48, but the LCM is actually 24.

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

LCM (Least Common Multiple) is the smallest number both divide into, while GCD (Greatest Common Divisor) is the largest number that divides both. For 3 and 4: LCM = 12, GCD = 1.

Q: Is there a faster way than listing multiples?

Yes! Use prime factorization: $3 = 3^1$ and $4 = 2^2$. Take the highest power of each prime: $2^2 \times 3^1 = 12$.

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

When adding or subtracting fractions, you need a common denominator. The LCM gives you the smallest one, making calculations easier and keeping numbers manageable!

Q: What if one number divides evenly into the other?

Then the larger number is the LCM! For example, with 3 and 12, since 12 ÷ 3 = 4 exactly, the LCM is 12. No need to list multiples!

LCM of Denominators with Small Integers

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

$\boxed 3~~~\boxed 4$

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Step-by-step written solution

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Understand the problem

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

$\boxed 3~~~\boxed 4$

Step-by-step solution

To find the least common multiple (LCM) of $3$ and $4$ , we list the multiples of each number until we find the smallest multiple they have in common.

Multiples of $3$ : $3, 6, 9, 12, 15, \ldots$

Multiples of $4$ : $4, 8, 12, 16, 20, \ldots$

The smallest common multiple is $12$ .

Final Answer

Key Points to Remember

Essential concepts to master this topic

Rule: LCM is the smallest positive number divisible by both denominators
Technique: List multiples until common one appears: 3, 6, 9, 12 and 4, 8, 12
Check: Verify 12 ÷ 3 = 4 and 12 ÷ 4 = 3 with no remainders ✓

Common Mistakes

Avoid these frequent errors

Multiplying the denominators instead of finding LCM
Don't just multiply 3 × 4 = 12 without checking if it's the smallest! Sometimes this works, but for numbers like 6 and 9, you'd get 54 instead of the correct LCM of 18. Always find the actual smallest common multiple by listing or using prime factorization.

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 multiply 3 × 4 = 12?

You can when the numbers are coprime (share no common factors), like 3 and 4! But this shortcut doesn't always work. For numbers like 6 and 8, multiplying gives 48, but the LCM is actually 24.

What's the difference between LCM and GCD?

LCM (Least Common Multiple) is the smallest number both divide into, while GCD (Greatest Common Divisor) is the largest number that divides both. For 3 and 4: LCM = 12, GCD = 1.

Is there a faster way than listing multiples?

Yes! Use prime factorization: $3 = 3^1$ and $4 = 2^2$ . Take the highest power of each prime: $2^2 \times 3^1 = 12$ .

Why do I need to find LCM of denominators?

When adding or subtracting fractions, you need a common denominator. The LCM gives you the smallest one, making calculations easier and keeping numbers manageable!

What if one number divides evenly into the other?

Then the larger number is the LCM! For example, with 3 and 12, since 12 ÷ 3 = 4 exactly, the LCM is 12. No need to list multiples!

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