5. (50 points) Vasya thought of a natural number not greater than 100. Fedya asks Vasya questions of the form “What is the quotient of the division of the thought number by $m$?”, where $m$ is a natural number not less than 2, and Fedya chooses it himself. What is the minimum number of such questions Fedya can guarantee to determine the number $B a$ that Vasya thought of?
Answer: 2.

Solution. Let Vasya think of a number $V$. Then $V=k \cdot m+d$, where $d$ can take values from 0 to $m-1$ inclusive, and Vasya's answer to Fyodor's question will be the number $k$. Since by the condition $m>1$, there are at least two variants for the remainder $d$ (the exception is the case when Vasya thought of the number 100 and Fyodor asks about the incomplete quotient from dividing by 100). Therefore, we cannot always determine the thought number with one question.

We will show how to do this in 2 questions. The first question from Fyodor should be: "What is the quotient of the division of the thought number by 2?". Let Vasya answer: "This quotient is $k$". Then Vasya either thought of the number $2 k$ or $2 k+1$. Then the second question

from Fyodor should be "What is the quotient of the division of the thought number by $2 k+1$?". In the case where Vasya thought of the number $2 k$, the answer will be 0, and if he thought of the number $2 k+1-$ the answer will be 1.