To find the smallest number that is a multiple of all the numbers 2, 3, 4, 5, 6, 8, and 9, we need to find the least common multiple (LCM) of these numbers.
First, we can list the prime factors of each number:
- 2: 2
- 3: 3
- 4: 2 x 2
- 5: 5
- 6: 2 x 3
- 8: 2 x 2 x 2
- 9: 3 x 3
Next, we can identify the highest power of each prime factor that appears in any of the numbers:
- 2: 2 x 2 x 2 = 8
- 3: 3 x 3 = 9
- 5: 5
So the LCM of 2, 3, 4, 5, 6, 8, and 9 is:
LCM = 2 x 2 x 2 x 3 x 3 x 5
LCM = 360
Therefore, the smallest number that is a multiple of all the numbers 2, 3, 4, 5, 6, 8, and 9 is 360.