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To find the greatest common divisor (GCD) and the least common multiple (LCM) of [tex]\( r = 3^3 \cdot 7^1 \cdot 11^2 \)[/tex] and [tex]\( s = 3^2 \cdot 7^2 \cdot 11^1 \)[/tex], we use the prime factorization method.
First, let's restate the prime factorizations of [tex]\( r \)[/tex] and [tex]\( s \)[/tex]:
- [tex]\( r = 3^3 \cdot 7^1 \cdot 11^2 \)[/tex]
- [tex]\( s = 3^2 \cdot 7^2 \cdot 11^1 \)[/tex]
### Finding the Greatest Common Divisor (GCD)
To calculate the GCD, we take each prime factor that appears in both factorizations and raise it to the lowest power found in the two numbers.
Let's break it down by each prime factor:
1. For the prime factor 3:
- The power of 3 in [tex]\( r \)[/tex] is [tex]\( 3 \)[/tex].
- The power of 3 in [tex]\( s \)[/tex] is [tex]\( 2 \)[/tex].
- Take the minimum of these powers: [tex]\( \min(3, 2) = 2 \)[/tex].
- So, the contribution to the GCD from the prime factor 3 is [tex]\( 3^2 \)[/tex].
2. For the prime factor 7:
- The power of 7 in [tex]\( r \)[/tex] is [tex]\( 1 \)[/tex].
- The power of 7 in [tex]\( s \)[/tex] is [tex]\( 2 \)[/tex].
- Take the minimum of these powers: [tex]\( \min(1, 2) = 1 \)[/tex].
- So, the contribution to the GCD from the prime factor 7 is [tex]\( 7^1 \)[/tex].
3. For the prime factor 11:
- The power of 11 in [tex]\( r \)[/tex] is [tex]\( 2 \)[/tex].
- The power of 11 in [tex]\( s \)[/tex] is [tex]\( 1 \)[/tex].
- Take the minimum of these powers: [tex]\( \min(2, 1) = 1 \)[/tex].
- So, the contribution to the GCD from the prime factor 11 is [tex]\( 11^1 \)[/tex].
Combining these, we get:
[tex]\[ \text{GCD} = 3^2 \cdot 7^1 \cdot 11^1 = 9 \cdot 7 \cdot 11 \][/tex]
Now, let's calculate this:
[tex]\[ 9 \cdot 7 = 63 \][/tex]
[tex]\[ 63 \cdot 11 = 693 \][/tex]
So, the GCD is:
[tex]\[ \boxed{693} \][/tex]
### Part 2: Finding the Least Common Multiple (LCM)
To calculate the LCM, we take each prime factor that appears in the factorizations and raise it to the highest power found in the two numbers.
Let's break it down by each prime factor:
1. For the prime factor 3:
- The power of 3 in [tex]\( r \)[/tex] is [tex]\( 3 \)[/tex].
- The power of 3 in [tex]\( s \)[/tex] is [tex]\( 2 \)[/tex].
- Take the maximum of these powers: [tex]\( \max(3, 2) = 3 \)[/tex].
- So, the contribution to the LCM from the prime factor 3 is [tex]\( 3^3 \)[/tex].
2. For the prime factor 7:
- The power of 7 in [tex]\( r \)[/tex] is [tex]\( 1 \)[/tex].
- The power of 7 in [tex]\( s \)[/tex] is [tex]\( 2 \)[/tex].
- Take the maximum of these powers: [tex]\( \max(1, 2) = 2 \)[/tex].
- So, the contribution to the LCM from the prime factor 7 is [tex]\( 7^2 \)[/tex].
3. For the prime factor 11:
- The power of 11 in [tex]\( r \)[/tex] is [tex]\( 2 \)[/tex].
- The power of 11 in [tex]\( s \)[/tex] is [tex]\( 1 \)[/tex].
- Take the maximum of these powers: [tex]\( \max(2, 1) = 2 \)[/tex].
- So, the contribution to the LCM from the prime factor 11 is [tex]\( 11^2 \)[/tex].
Combining these, we get:
[tex]\[ \text{LCM} = 3^3 \cdot 7^2 \cdot 11^2 \][/tex]
Now, let's calculate this:
[tex]\[ 3^3 = 27 \][/tex]
[tex]\[ 7^2 = 49 \][/tex]
[tex]\[ 11^2 = 121 \][/tex]
So:
[tex]\[ 27 \cdot 49 = 1323 \][/tex]
[tex]\[ 1323 \cdot 121 = 160083 \][/tex]
Thus, the LCM is:
[tex]\[ \boxed{160083} \][/tex]
To summarize, using the prime factorization method:
- The GCD is [tex]\( 693 \)[/tex].
- The LCM is [tex]\( 160083 \)[/tex].
First, let's restate the prime factorizations of [tex]\( r \)[/tex] and [tex]\( s \)[/tex]:
- [tex]\( r = 3^3 \cdot 7^1 \cdot 11^2 \)[/tex]
- [tex]\( s = 3^2 \cdot 7^2 \cdot 11^1 \)[/tex]
### Finding the Greatest Common Divisor (GCD)
To calculate the GCD, we take each prime factor that appears in both factorizations and raise it to the lowest power found in the two numbers.
Let's break it down by each prime factor:
1. For the prime factor 3:
- The power of 3 in [tex]\( r \)[/tex] is [tex]\( 3 \)[/tex].
- The power of 3 in [tex]\( s \)[/tex] is [tex]\( 2 \)[/tex].
- Take the minimum of these powers: [tex]\( \min(3, 2) = 2 \)[/tex].
- So, the contribution to the GCD from the prime factor 3 is [tex]\( 3^2 \)[/tex].
2. For the prime factor 7:
- The power of 7 in [tex]\( r \)[/tex] is [tex]\( 1 \)[/tex].
- The power of 7 in [tex]\( s \)[/tex] is [tex]\( 2 \)[/tex].
- Take the minimum of these powers: [tex]\( \min(1, 2) = 1 \)[/tex].
- So, the contribution to the GCD from the prime factor 7 is [tex]\( 7^1 \)[/tex].
3. For the prime factor 11:
- The power of 11 in [tex]\( r \)[/tex] is [tex]\( 2 \)[/tex].
- The power of 11 in [tex]\( s \)[/tex] is [tex]\( 1 \)[/tex].
- Take the minimum of these powers: [tex]\( \min(2, 1) = 1 \)[/tex].
- So, the contribution to the GCD from the prime factor 11 is [tex]\( 11^1 \)[/tex].
Combining these, we get:
[tex]\[ \text{GCD} = 3^2 \cdot 7^1 \cdot 11^1 = 9 \cdot 7 \cdot 11 \][/tex]
Now, let's calculate this:
[tex]\[ 9 \cdot 7 = 63 \][/tex]
[tex]\[ 63 \cdot 11 = 693 \][/tex]
So, the GCD is:
[tex]\[ \boxed{693} \][/tex]
### Part 2: Finding the Least Common Multiple (LCM)
To calculate the LCM, we take each prime factor that appears in the factorizations and raise it to the highest power found in the two numbers.
Let's break it down by each prime factor:
1. For the prime factor 3:
- The power of 3 in [tex]\( r \)[/tex] is [tex]\( 3 \)[/tex].
- The power of 3 in [tex]\( s \)[/tex] is [tex]\( 2 \)[/tex].
- Take the maximum of these powers: [tex]\( \max(3, 2) = 3 \)[/tex].
- So, the contribution to the LCM from the prime factor 3 is [tex]\( 3^3 \)[/tex].
2. For the prime factor 7:
- The power of 7 in [tex]\( r \)[/tex] is [tex]\( 1 \)[/tex].
- The power of 7 in [tex]\( s \)[/tex] is [tex]\( 2 \)[/tex].
- Take the maximum of these powers: [tex]\( \max(1, 2) = 2 \)[/tex].
- So, the contribution to the LCM from the prime factor 7 is [tex]\( 7^2 \)[/tex].
3. For the prime factor 11:
- The power of 11 in [tex]\( r \)[/tex] is [tex]\( 2 \)[/tex].
- The power of 11 in [tex]\( s \)[/tex] is [tex]\( 1 \)[/tex].
- Take the maximum of these powers: [tex]\( \max(2, 1) = 2 \)[/tex].
- So, the contribution to the LCM from the prime factor 11 is [tex]\( 11^2 \)[/tex].
Combining these, we get:
[tex]\[ \text{LCM} = 3^3 \cdot 7^2 \cdot 11^2 \][/tex]
Now, let's calculate this:
[tex]\[ 3^3 = 27 \][/tex]
[tex]\[ 7^2 = 49 \][/tex]
[tex]\[ 11^2 = 121 \][/tex]
So:
[tex]\[ 27 \cdot 49 = 1323 \][/tex]
[tex]\[ 1323 \cdot 121 = 160083 \][/tex]
Thus, the LCM is:
[tex]\[ \boxed{160083} \][/tex]
To summarize, using the prime factorization method:
- The GCD is [tex]\( 693 \)[/tex].
- The LCM is [tex]\( 160083 \)[/tex].
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