What is the highest power of 2 in 100! ?

To find the highest power of 2 in 100!, we need to count the number of times 2 appears as a factor in the numbers from 1 to 100.

1. First, divide 100 by 2. This gives us 50. This means there are 50 numbers that are divisible by 2 in the range 1 to 100.

2. But some numbers are divisible by 2 more than once. For example, 4 is 22, 8 is 22*2, and so on. So, we need to count these extra factors of 2. We do this by dividing 100 by 4 (which is 2^2). This gives us 25.

3. We repeat this process for 2^3 = 8, 2^4 = 16, 2^5 = 32, and 2^6 = 64. We get 12, 6, 3, and 1 respectively.

4. We don't need to go beyond 2^6 because 2^7 = 128 is greater than 100.

5. Finally, we add up all these counts: 50 + 25 + 12 + 6 + 3 + 1 = 97.

So, the highest power of 2 in 100! is 97.