If n is a natural number less than 100 and k is any whole number, for how many values of n is n2 = 24k + 1?

To solve the problem, we need to find the values of $n$ such that $n^2 = 24k + 1$ for some whole number $k$. Here are the steps:

1. Rewrite the equation: $n^2 = 24k + 1$ This can be rearranged to: $n^2 - 1 = 24k$ $(n - 1)(n + 1) = 24k$

2. Analyze the factors: Since $(n - 1)(n + 1)$ must be a multiple of 24, we need to consider the properties of these factors. Note that $n - 1$ and $n + 1$ are two consecutive even numbers (since their difference is 2).

3. Factor pairs of 24: The factor pairs of 24 are: $(1, 24), (2, 12), (3, 8), (4, 6)$ We need to check which pairs can be consecutive even numbers.

4. Check each pair:

   • For $(1, 24)$: Not possible since they are not consecutive.
   • For $(2, 12)$: Not possible since they are not consecutive.
   • For $(3, 8)$: Not possible since they are not consecutive.
   • For $(4, 6)$: Possible since they are consecutive even numbers.

5. Find $n$ for valid pairs:

   • For $(4, 6)$: $n - 1 = 4$ $n + 1 = 6$ Solving these, we get: $n = 5$

6. Check other multiples of 24: We need to consider multiples of 24, such as 48, 72, etc., and check if they can be written as products of two consecutive even numbers.

   • For 48: $(6, 8)$ is a valid pair. $n - 1 = 6$ $n + 1 = 8$ Solving these, we get: $n = 7$

   • For 72: $(8, 9)$ is not a valid pair since they are not both even.

   • For 96: $(10, 12)$ is a valid pair. $n - 1 = 10$ $n + 1 = 12$ Solving these, we get: $n = 11$

7. List all valid $n$ values: From the above steps, the valid values of $n$ are: $n = 5, 7, 11$

8. Count the values: There are 3 values of $n$ that satisfy the given equation.

Therefore, the number of values of $n$ is $\boxed{3}$.