Bob has picked positive integer $1<N<100$. Alice tells him some integer, and Bob replies with the remainder of division of this integer by $N$. What is the smallest number of integers which Alice should tell Bob to determine $N$ for sure?
To determine the smallest number of integers Alice should tell Bob to determine \( N \) for sure, we need to consider the information obtained from each guess and how it narrows down the possible values of \( N \).

1. **Initial Consideration**:
   - Alice needs to determine \( N \) where \( 1 < N < 100 \).
   - Each guess \( n \) provides a remainder \( r \) when divided by \( N \), i.e., \( r = n \mod N \).

2. **Information from Each Guess**:
   - If Alice guesses \( n \) and Bob replies with \( r \):
     - If \( r = n \), then \( N > n \).
     - If \( r < n \), then \( N \) must be a divisor of \( n - r \).

3. **Strategy to Minimize Guesses**:
   - To minimize the number of guesses, Alice should choose \( n \) such that the number of possible values for \( N \) is reduced as much as possible with each guess.
   - The best strategy is to use guesses that are powers of 2, as they provide the maximum reduction in the number of possible values for \( N \).

4. **Detailed Steps**:
   - **First Guess**: Let \( n_1 = 64 \). 
     - If \( r_1 = 64 \), then \( N > 64 \). This reduces the possible values of \( N \) to \( 65, 66, \ldots, 99 \) (35 possibilities).
     - If \( r_1 < 64 \), then \( N \) must be a divisor of \( 64 - r_1 \). This provides multiple possible values for \( N \) but significantly reduces the range.
   - **Second Guess**: Let \( n_2 = 32 \).
     - If \( r_2 = 32 \), then \( N > 32 \). This further reduces the possible values of \( N \).
     - If \( r_2 < 32 \), then \( N \) must be a divisor of \( 32 - r_2 \).
   - **Third Guess**: Let \( n_3 = 16 \).
     - If \( r_3 = 16 \), then \( N > 16 \). This further narrows down the possible values of \( N \).
     - If \( r_3 < 16 \), then \( N \) must be a divisor of \( 16 - r_3 \).
   - **Fourth Guess**: Let \( n_4 = 8 \).
     - If \( r_4 = 8 \), then \( N > 8 \). This further narrows down the possible values of \( N \).
     - If \( r_4 < 8 \), then \( N \) must be a divisor of \( 8 - r_4 \).
   - **Fifth Guess**: Let \( n_5 = 4 \).
     - If \( r_5 = 4 \), then \( N > 4 \). This further narrows down the possible values of \( N \).
     - If \( r_5 < 4 \), then \( N \) must be a divisor of \( 4 - r_5 \).
   - **Sixth Guess**: Let \( n_6 = 2 \).
     - If \( r_6 = 2 \), then \( N > 2 \). This further narrows down the possible values of \( N \).
     - If \( r_6 < 2 \), then \( N \) must be a divisor of \( 2 - r_6 \).

5. **Conclusion**:
   - By using the above strategy, Alice can determine \( N \) with a maximum of 6 guesses. Each guess significantly reduces the number of possible values for \( N \).

The final answer is \(\boxed{6}\).