Let 3^a be the highest power of 3 that divides 1000! . What is a ?
To find the highest power of 3 that divides 1000!, we need to determine the number of multiples of 3 present in the factorial. We can do this by counting the number of times 3 appears as a factor, along with the number of times any multiple of 3 appears (such as 6, 9, 12, etc.).
To calculate how many times 3 appears as a factor, we divide 1000 by 3, which gives us 333. This means there are 333 multiples of 3 from 1 to 1000.
However, this count does not account for multiples of 3 squared or higher powers of 3. To include those, we need to divide 1000 by 3 again: (1000 ÷ 3^2 = 111). This gives us 111 more multiples of 3 squared.
Now, we repeat this process to account for multiples of 3 cubed: (1000 ÷ 3^3 = 37). This adds 37 more multiples of 3 cubed.
Since 3^4 (81) is greater than 1000, we can stop here. Adding up the counts, we have 333 + 111 + 37 = 481.
Therefore, the highest power of 3 that divides 1000! is 3^481. Hence, a = 481.