Step 1: Problem Definition:
The objective is to identify the smallest integer that is a multiple of 8, 9, and 10. This value is equivalent to the Least Common Multiple (LCM) of these three numbers.
Step 2: Prime Factorization:
The prime factorization for each number is as follows:
- $8 = 2^3$
- $9 = 3^2$
- $10 = 2 \times 5$
Step 3: LCM Calculation:
To compute the LCM, we select the highest exponent for each unique prime factor present across all factorizations.
- The maximum power of 2 is $2^3$.
- The maximum power of 3 is $3^2$.
- The maximum power of 5 is $5^1$.
The LCM is calculated as:
\[
\text{LCM} = 2^3 \times 3^2 \times 5 = 8 \times 9 \times 5 = 360
\]
Step 4: Final Answer:
The smallest number divisible by 8, 9, and 10 is 360.