Find the highest power of 3 in 100!

To find the highest power of 3 in 100!, we need to count the number of times 3 appears as a factor in the numbers from 1 to 100.

Step 1: Divide 100 by 3, we get 33. This means that there are 33 numbers that are divisible by 3 in the range 1 to 100.

Step 2: But some numbers may have more than one 3 as a factor. For example, 9 = 3^2, 27 = 3^3, etc. To account for these, we divide 100 by 3^2 = 9. We get 11. So, there are 11 numbers that have at least two 3s as factors.

Step 3: Continuing this process, we divide 100 by 3^3 = 27. We get 3. So, there are 3 numbers that have at least three 3s as factors.

Step 4: If we divide 100 by 3^4 = 81, we get 1. So, there is 1 number that has at least four 3s as factors.

Step 5: If we try to divide 100 by 3^5 = 243, we get a number less than 1. So, there are no numbers that have at least five 3s as factors.

Adding up all these counts, we get 33 + 11 + 3 + 1 = 48. So, the highest power of 3 in 100! is 3^48.