Find the LCM of Denominators: 2, 7, and 9 - Step-by-Step Solution

Q: How do I find prime factorization of larger numbers?

Start with the smallest prime (2) and keep dividing: If even, divide by 2If odd, try 3, then 5, then 7, etc.Stop when you reach 1

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

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

Q: Why do we need the LCM when adding fractions?

To add fractions like $ \frac{1}{2} + \frac{1}{7} + \frac{1}{9} $, you need a common denominator. The LCM gives you the smallest possible common denominator, making calculations easier!

Prime Factorization with LCM Calculations

Given several denominators, what is their least common multiple?

$\boxed2~~~\boxed7 ~~~\boxed9$

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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?

$\boxed2~~~\boxed7 ~~~\boxed9$

Step-by-step solution

To find the least common multiple (LCM) of the denominators $2, 7, 9$ , we need to consider each prime factor of these numbers at their highest power:

$2$ : prime itself
$7$ : prime itself
$9 = 3^2$

Therefore, the LCM is:

$2 \times 7 \times 3^2 = 126$

So, the least common multiple of $2, 7, 9$ is $126$ .

Final Answer

126

Key Points to Remember

Essential concepts to master this topic

Prime Factorization: Break each number into prime factors first
Technique: Take highest power of each prime: $2^1 \times 7^1 \times 3^2 = 126$
Check: Verify 126 ÷ 2 = 63, 126 ÷ 7 = 18, 126 ÷ 9 = 14 ✓

Common Mistakes

Avoid these frequent errors

Adding denominators instead of using prime factorization
Don't just add 2 + 7 + 9 = 18! This gives a number that's not divisible by all denominators. Always find prime factorization first, then multiply the highest power of each prime factor.

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 2 × 7 × 9 = 126 happens to work here, but it doesn't always give the least common multiple. For example, with 4 and 6, multiplying gives 24, but the LCM is only 12!

What if one of the numbers is already a factor of another?

Great question! If one number divides evenly into another, the larger number is automatically the LCM. For example, with 3 and 9, the LCM is just 9 since 9 = 3 × 3.

How do I find prime factorization of larger numbers?

Start with the smallest prime (2) and keep dividing:

If even, divide by 2
If odd, try 3, then 5, then 7, etc.
Stop when you reach 1

What's the difference between LCM and GCD?

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

Why do we need the LCM when adding fractions?

To add fractions like $\frac{1}{2} + \frac{1}{7} + \frac{1}{9}$ , you need a common denominator. The LCM gives you the smallest possible common denominator, making calculations easier!

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