What power of 8 exactly divides 25!?

To find the power of 8 that exactly divides 25!, we first need to find the number of 2's in the prime factorization of 25!.

Step 1: Find the number of 2's in the prime factorization of 25! The number of 2's in the prime factorization of 25! can be found by the formula n/2 + n/4 + n/8 + n/16 + ..., where n is the number 25. This formula is used to find the number of times a prime number p divides n!.

So, the number of 2's in the prime factorization of 25! is 25/2 + 25/4 + 25/8 + 25/16 = 12 + 6 + 3 + 1 = 22.

Step 2: Find the power of 8 that exactly divides 25! Since 8 is 2^3, we need to find how many groups of 3 we can make from the 22 2's.

So, the power of 8 that exactly divides 25! is 22/3 = 7 (since we ignore the remainder).

Therefore, the power of 8 that exactly divides 25! is 7.