Solution:

To find the highest power of 7 in 100!, we need to find the number of factors of 7 in the prime factorization of 100!.

Prime Factorization of 100!:

100! = 2^97 × 3^48 × 5^24 × 7^16 × 11^9 × 13^7 × 17^5 × 19^5 × 23^4 × 29^3 × 31^3 × 37^2 × 41^2 × 43^2 × 47^2 × 53 × 59 × 61 × 67 × 71 × 73 × 79 × 83 × 89 × 97

The number of factors of 7 in the prime factorization of 100! is the same as the power of 7 in the prime factorization of 100!.

Therefore, we need to find the highest power of 7 in 100!.

Finding the highest power of 7 in 100!:

To find the highest power of 7 in 100!, we need to find the number of multiples of 7 in the numbers from 1 to 100.

We can find the number of multiples of 7 in the numbers from 1 to 100 by using the floor function.

The floor function of a number x, denoted by [x], is the largest integer that is less than or equal to x.

Therefore, the number of multiples of 7 in the numbers from 1 to 100 is given by: [100/7] = 14.

However, we have to consider that some of these multiples of 7 are also multiples of 49, which means that they have an extra factor of 7.

To find the number of multiples of 49 in the numbers from 1 to 100, we can use the same approach.

The number of multiples of 49 in the numbers from 1 to 100 is given by: [100/49] = 2.

Therefore, the total number of factors of 7 in the prime factorization of 100! is 14 + 2 = 16.

Hence, the highest power of 7 in 100! is 7^16.

Final Answer: Highest power of 7 in 100! = [100/7] [100/49] = 16.