Finding the LCM of 9 and 12
When we talk about Least Common Multiple (LCM) in math, we are simply referring to the smallest number that is a multiple of two or more numbers. In this case, we'll explore how to find the LCM of 9 and 12. Understanding how to calculate the LCM can make topics like fractions, ratios, and algebra much easier to tackle.
What Is LCM?
The Least Common Multiple of two or more numbers is the smallest number that is evenly divisible by each of the given numbers. It’s a handy concept, especially when dealing with fractions or when trying to synchronize different cyclic events.
Ways To Calculate LCM
There are several methods to calculate the LCM of numbers. Here are three popular methods:
- Division Method
- Listing Method
- Prime Factorization Method
Let’s break down these methods to find the LCM of 9 and 12.
Prime Factorization Method
When using the Prime Factorization Method, we first break down each number into its prime factors. Then, we take the highest power of all prime numbers that appear in these factorizations.
- Factorize 9: 9 = 3 x 3 (or 3^2)
- Factorize 12: 12 = 2 x 2 x 3 (or 2^2 x 3)
Now, for each prime number, take the highest power:
- For 2: The highest power is 2^2
- For 3: The highest power is 3^2
Multiply these together:
LCM = 2^2 x 3^2 = 4 x 9 = 36
So, the LCM of 9 and 12 using the Prime Factorization Method is 36.
Division Method
The Division Method involves dividing the numbers by common prime factors until we are left with 1.
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Write the numbers side by side and start dividing by the smallest prime number that can divide at least one of the numbers.
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Continue dividing until only 1s remain.
2 | 9, 12
3 | 9, 6
3 | 3, 2
2 | 1, 2
1 | 1, 1
Multiply all the prime factors:
LCM = 2 x 3 x 3 x 2 = 36
So, the LCM of 9 and 12 using the Division Method is also 36.
Listing Method
In the Listing Method, we list the multiples of each number until we find the smallest common multiple.
- Multiples of 9: 9, 18, 27, 36, 45, 54…
- Multiples of 12: 12, 24, 36, 48, 60…
Both lists first share the number 36.
So, the LCM of 9 and 12 using the Listing Method is 36.
LCM Formula
An easy way to remember how to find the LCM using prime factors is with the following formula:
LCM = Product of the highest powers of all prime factors involved
Conclusion
Calculating the Least Common Multiple (LCM) of numbers like 9 and 12 can be done in several ways: the Prime Factorization Method, the Division Method, and the Listing Method. Each method has its own steps, but they all lead to the same result. The LCM of 9 and 12 is 36, making it the smallest number that both 9 and 12 can divide into evenly.
FAQs
Q: What does LCM stand for?
A: LCM stands for Least Common Multiple, which is the smallest multiple that is evenly divisible by two or more numbers.
Q: Why do we need to find the LCM?
A: Finding the LCM is useful in solving problems involving adding, subtracting, or comparing fractions, and also in finding common denominators.
Q: Is there a quickest method to find the LCM?
A: The quickest method depends on the situation. The Prime Factorization Method is great for larger numbers, whereas the Listing Method is useful for smaller numbers or when you need a quick answer.
Q: Can the LCM of two numbers be one of the numbers?
A: Yes, this happens when one number is a multiple of the other. For example, the LCM of 5 and 10 is 10.
