GCD And LCM: Calculation Examples For A And B

Hey guys! Let's dive into finding the Greatest Common Divisor (GCD) and the Least Common Multiple (LCM) of two numbers. It might sound a bit intimidating, but trust me, it's super useful and kinda fun once you get the hang of it. We're going to break it down step-by-step with some examples. So, grab your thinking caps, and let's get started!

Understanding GCD and LCM

Before we jump into the calculations, let's quickly recap what GCD and LCM actually mean.

• GCD (Greatest Common Divisor): The GCD of two or more numbers is the largest positive integer that divides each of the numbers without leaving a remainder. Basically, it's the biggest number that can evenly divide both numbers. Think of it as the biggest shared factor.
• LCM (Least Common Multiple): The LCM of two or more numbers is the smallest positive integer that is divisible by each of the numbers. It's the smallest number that both numbers can divide into evenly. Think of it as the smallest shared multiple.

Knowing these definitions is crucial because they guide our approach when we're solving problems. We need to understand what we're actually trying to find to solve it effectively. We will use prime factorization to find GCD and LCM. Prime factorization will allow us to break down the numbers into their prime factors and make it easier to identify common factors and multiples.

Example Problem a: Finding GCD and LCM

Okay, let's tackle our first example. We're given two numbers in their prime factorized forms:

• $a = 2^3 \cdot 3^1 \cdot 5^2$
• $b = 2^2 \cdot 3^3 \cdot 5^1$

Finding the GCD

To find the GCD, we need to identify the common prime factors and take the lowest power of each. This is because the GCD can only include factors that are present in both numbers, and we're looking for the greatest common divisor, so we take the lowest power to ensure it divides both numbers.

1. Identify common prime factors: Both a and b have 2, 3, and 5 as prime factors.
2. Take the lowest power of each common prime factor:
   • For 2, the lowest power is $2^2$.
   • For 3, the lowest power is $3^1$.
   • For 5, the lowest power is $5^1$.
3. Multiply these lowest powers together:
   • $GCD(a, b) = 2^2 \cdot 3^1 \cdot 5^1 = 4 \cdot 3 \cdot 5 = 60$

So, the GCD of a and b is 60. This means that 60 is the largest number that divides both $2^3 \cdot 3^1 \cdot 5^2$ and $2^2 \cdot 3^3 \cdot 5^1$ without leaving a remainder. Pretty neat, huh?

Finding the LCM

Now, let's find the LCM. For the LCM, we'll use a similar approach, but this time, we take the highest power of each prime factor present in either number. This ensures that the LCM is divisible by both numbers.

1. Identify all prime factors: The prime factors in a and b are 2, 3, and 5.
2. Take the highest power of each prime factor:
   • For 2, the highest power is $2^3$.
   • For 3, the highest power is $3^3$.
   • For 5, the highest power is $5^2$.
3. Multiply these highest powers together:
   • $LCM(a, b) = 2^3 \cdot 3^3 \cdot 5^2 = 8 \cdot 27 \cdot 25 = 5400$

Therefore, the LCM of a and b is 5400. This means that 5400 is the smallest number that is divisible by both $2^3 \cdot 3^1 \cdot 5^2$ and $2^2 \cdot 3^3 \cdot 5^1$. We have successfully found both the GCD and the LCM for this example!

Example Problem b: Finding GCD and LCM

Let's move on to our second example to solidify our understanding. This time, we have:

• $a = 3^3 \cdot 5^1 \cdot 7^2$
• $b = 2^2 \cdot 3^1 \cdot 5^1$

Finding the GCD

Remember, for the GCD, we look for the common prime factors and take the lowest power of each.

1. Identify common prime factors: Both a and b have 3 and 5 as prime factors.
2. Take the lowest power of each common prime factor:
   • For 3, the lowest power is $3^1$.
   • For 5, the lowest power is $5^1$.
3. Multiply these lowest powers together:
   • $GCD(a, b) = 3^1 \cdot 5^1 = 3 \cdot 5 = 15$

So, the GCD of a and b is 15. This means 15 is the largest number that can divide both $3^3 \cdot 5^1 \cdot 7^2$ and $2^2 \cdot 3^1 \cdot 5^1$ without any remainder.

Finding the LCM

For the LCM, we take the highest power of each prime factor present in either number.

1. Identify all prime factors: The prime factors in a and b are 2, 3, 5, and 7.
2. Take the highest power of each prime factor:
   • For 2, the highest power is $2^2$.
   • For 3, the highest power is $3^3$.
   • For 5, the highest power is $5^1$.
   • For 7, the highest power is $7^2$.
3. Multiply these highest powers together:
   • $LCM(a, b) = 2^2 \cdot 3^3 \cdot 5^1 \cdot 7^2 = 4 \cdot 27 \cdot 5 \cdot 49 = 26460$

Thus, the LCM of a and b is 26460. This is the smallest number that is divisible by both $3^3 \cdot 5^1 \cdot 7^2$ and $2^2 \cdot 3^1 \cdot 5^1$. Great job, we've solved another one!

Key Takeaways for GCD and LCM

Alright, let's recap the main points to remember when finding the GCD and LCM:

• GCD:
  • Find the common prime factors. The prime factors are crucial because they form the building blocks of the numbers. You absolutely must break down the numbers into their prime factors to proceed correctly.
  • Take the lowest power of each common prime factor. Think of it as selecting the smallest common ingredient.
  • Multiply these lowest powers together to get the GCD. This gives you the biggest number that divides both given numbers.
• LCM:
  • Find all prime factors present in either number. Consider every prime that appears in any of the numbers. Nothing should be left out.
  • Take the highest power of each prime factor. Imagine you are combining recipes and need enough of each ingredient; you take the largest amount required.
  • Multiply these highest powers together to get the LCM. This produces the smallest number divisible by both original numbers.

Why are GCD and LCM Important?

You might be wondering,