Problem Statement: Determine the smallest integer divisible by 8, 9, and 10. This is equivalent to finding the Least Common Multiple (LCM) of 8, 9, and 10.
Prime Factorization:
- \( 8 = 2^3 \)
- \( 9 = 3^2 \)
- \( 10 = 2 \times 5 \)
LCM Calculation: The LCM is formed by taking the highest power of each prime factor present in any of the numbers.
- Highest power of 2: \( 2^3 \)
- Highest power of 3: \( 3^2 \)
- Highest power of 5: \( 5^1 \)
Therefore, the LCM is calculated as:\[\text{LCM} = 2^3 \times 3^2 \times 5 = 8 \times 9 \times 5 = 72 \times 5 = 360\]
Result: The smallest number divisible by 8, 9, and 10 is \( \boxed{360} \).