Find the LCM of Denominators: 3, 9, and 12

Q: How do I know I found the right prime factorization?

Check your work by multiplying the prime factors back together. For example: $ 2^2 \times 3 = 4 \times 3 = 12 $ ✓. If you get the original number, your factorization is correct!

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

LCM (Least Common Multiple) is the smallest number that all given numbers divide into evenly. GCD (Greatest Common Divisor) is the largest number that divides evenly into all given numbers.

LCM with Prime Factor Method

Identify the least common multiple of these denominators:

$\boxed{3}~~~\boxed{9} ~~~\boxed{12}$

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Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Identify the least common multiple of these denominators:

$\boxed{3}~~~\boxed{9} ~~~\boxed{12}$

Step-by-step solution

To find the least common multiple (LCM) of $3$ , $9$ , and $12$ , begin by finding their prime factorizations:

$3 = 3$

$9 = 3^2$

$12 = 2^2 \, \times \, 3$

The LCM is calculated by taking the highest power of each prime present:

Max of $2$ is $2^2$ and of $3$ is $3^2$ .

Thus, LCM is $2^2 \, \times \, 3^2 = 4 \, \times \, 9 = 36$ .

Final Answer

Key Points to Remember

Essential concepts to master this topic

Rule: Find LCM using highest power of each prime factor
Technique: For 3, 9, 12: use $2^2 \times 3^2 = 36$
Check: Verify 36 divides evenly by all three numbers: 36÷3=12, 36÷9=4, 36÷12=3 ✓

Common Mistakes

Avoid these frequent errors

Multiplying all numbers together instead of using prime factorization
Don't just multiply 3×9×12 = 324! This gives you a common multiple but not the LEAST common multiple. Always find the highest power of each prime factor to get the smallest possible 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 multiply all the numbers together?

Multiplying all numbers gives you a common multiple, but not the least one! For 3, 9, and 12, multiplying gives 324, but the LCM is only 36. Using prime factorization finds the smallest number that works.

What if one number is already a multiple of another?

Great observation! Since 9 = 3², the number 9 already contains all the prime factors of 3. So when finding the LCM, you only need the highest power of each prime factor.

How do I know I found the right prime factorization?

Check your work by multiplying the prime factors back together. For example: $2^2 \times 3 = 4 \times 3 = 12$ ✓. If you get the original number, your factorization is correct!

What's the difference between LCM and GCD?

LCM (Least Common Multiple) is the smallest number that all given numbers divide into evenly. GCD (Greatest Common Divisor) is the largest number that divides evenly into all given numbers.

Can I use this method for fractions too?

Absolutely! When adding or subtracting fractions, you need a common denominator. Finding the LCM of the denominators gives you the least common denominator (LCD) to use.

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