Question

Find the smallest number that is divisible by each of 8, 9, and 10.

Answer 1

Step 1: Understand the problem:
We are asked to find the smallest number that is divisible by 8, 9, and 10. This is the least common multiple (LCM) of 8, 9, and 10.

Step 2: Find the prime factorization of each number:
- The prime factorization of 8 is \( 8 = 2^3 \). - The prime factorization of 9 is \( 9 = 3^2 \). - The prime factorization of 10 is \( 10 = 2 \times 5 \).

Step 3: Use the LCM formula:
The LCM of two or more numbers is found by taking the highest powers of all the primes that appear in the factorizations of the numbers.
- The highest power of 2 is \( 2^3 \) (from 8). - The highest power of 3 is \( 3^2 \) (from 9). - The highest power of 5 is \( 5^1 \) (from 10).

Thus, the LCM of 8, 9, and 10 is: \[ \text{LCM} = 2^3 \times 3^2 \times 5 = 8 \times 9 \times 5 = 72 \times 5 = 360 \]

Step 4: Conclusion:
The smallest number that is divisible by 8, 9, and 10 is \( \boxed{360} \).