To find the least common multiple (LCM) of 8 and 12, you can use the prime factorization method or the formula involving the greatest common divisor (GCD). Here's how to do it using both methods:
Method 1: Prime Factorization
-
Find the prime factorization of both numbers:
- 8 = \(2^3\)
- 12 = \(2^2 \times 3^1\)
-
Take the highest power of each prime factor:
- For 2: the highest power is \(2^3\)
- For 3: the highest power is \(3^1\)
-
Multiply these together to find the LCM: \[ \text{LCM} = 2^3 \times 3^1 = 8 \times 3 = 24 \]
Method 2: Using GCD
-
First, find the GCD of 8 and 12, which is 4.
-
Use the LCM formula: \[ \text{LCM}(a, b) = \frac{|a \times b|}{\text{GCD}(a, b)} \] where \(a = 8\) and \(b = 12\).
-
Substitute the values: \[ \text{LCM}(8, 12) = \frac{8 \times 12}{4} = \frac{96}{4} = 24 \]
In both methods, we find that the LCM of 8 and 12 is 24.