Work out the difference of the lcm of 12 and 24 and the gcd of 36 and 54
To find the LCM of 12 and 24, we need to prime factorize both numbers.
12 can be prime factorized as 2^2 * 3
24 can be prime factorized as 2^3 * 3
To find the LCM, we need to take the highest power of each prime factor.
The LCM of 12 and 24 is therefore 2^3 * 3 = 24.
To find the GCD of 36 and 54, we prime factorize both numbers.
36 can be prime factorized as 2^2 * 3^2
54 can be prime factorized as 2 * 3^3
To find the GCD, we need to take the smallest power of each prime factor.
The GCD of 36 and 54 is therefore 2^2 * 3 = 12.
The difference between the LCM of 12 and 24 (24) and the GCD of 36 and 54 (12) is 24 - 12 = 12.
To find the difference between the LCM of 12 and 24 and the GCD of 36 and 54, let's break down the steps:
Step 1: Find the LCM of 12 and 24
To find the LCM (Least Common Multiple) of 12 and 24, we need to find the smallest number that is divisible by both 12 and 24.
Prime factorization of 12: 2^2 * 3^1
Prime factorization of 24: 2^3 * 3^1
Now, we take the highest power of each prime factor that appears in either number.
LCM = 2^3 * 3^1 = 8 * 3 = 24
So, the LCM of 12 and 24 is 24.
Step 2: Find the GCD of 36 and 54
To find the GCD (Greatest Common Divisor) of 36 and 54, we need to find the largest number that divides both 36 and 54 without any remainder.
Prime factorization of 36: 2^2 * 3^2
Prime factorization of 54: 2^1 * 3^3
Again, we take the highest power of each prime factor that appears in either number.
GCD = 2^1 * 3^2 = 2 * 9 = 18
So, the GCD of 36 and 54 is 18.
Step 3: Calculate the difference
Now, subtract the GCD from the LCM to find the difference.
Difference = LCM - GCD
Difference = 24 - 18
Difference = 6
Therefore, the difference between the LCM of 12 and 24 and the GCD of 36 and 54 is 6.
To work out the difference between the LCM (Least Common Multiple) of 12 and 24 and the GCD (Greatest Common Divisor) of 36 and 54, follow these steps:
Step 1: Find the LCM of 12 and 24
- Start by listing the prime factors of each number: 12 = 2^2 * 3 and 24 = 2^3 * 3.
- The LCM is found by taking the highest power of each prime factor: LCM(12, 24) = 2^3 * 3 = 8 * 3 = 24.
Step 2: Find the GCD of 36 and 54
- Start by listing the prime factors of each number: 36 = 2^2 * 3^2 and 54 = 2 * 3^3.
- The GCD is found by taking the lowest power of each common prime factor: GCD(36, 54) = 2^1 * 3^2 = 2 * 9 = 18.
Step 3: Calculate the difference
- The difference is given by subtracting the GCD from the LCM: LCM(12, 24) - GCD(36, 54) = 24 - 18 = 6.
Therefore, the difference between the LCM of 12 and 24 and the GCD of 36 and 54 is 6.