Step 1: Prime Factorization
Determine the prime factors for 24, 36, and 60.
Prime Factors of 24:
\[24 = 2 \times 12 = 2 \times 2 \times 6 = 2 \times 2 \times 2 \times 3 = 2^3 \times 3\]
Prime Factors of 36:
\[36 = 2 \times 18 = 2 \times 2 \times 9 = 2 \times 2 \times 3 \times 3 = 2^2 \times 3^2\]
Prime Factors of 60:
\[60 = 2 \times 30 = 2 \times 2 \times 15 = 2 \times 2 \times 3 \times 5 = 2^2 \times 3 \times 5\]
Step 2: Calculate LCM from Prime Factors
The LCM is derived by selecting the highest power of each unique prime factor present across all numbers.
- Highest power of 2: \( 2^3 \) (from 24).
- Highest power of 3: \( 3^2 \) (from 36).
- Highest power of 5: \( 5^1 \) (from 60).
Step 3: LCM of 24, 36, and 60
Multiply the highest powers of all identified prime factors to find the LCM:
\[\text{LCM} = 2^3 \times 3^2 \times 5\]
Conclusion:
The LCM of 24, 36, and 60, expressed using prime factors, is \( 2^3 \times 3^2 \times 5 \).