6.55 21 people participate in an exam, the test paper contains 15 true/false questions. It is known that any two people have at least 1 question that they both answered correctly. What is the minimum number of people who answered the question with the most correct answers? Explain your reasoning.
[Solution] We call two people who answer the same question correctly a "common question group". Since any two people have at least 1 question in common, there are at least $C_{21}^{2}=210$ common question groups.

If no more than 5 people answer each question correctly, then each question can result in at most 10 common question groups, making the total number of common question groups no more than 150, which is a contradiction. If no more than 6 people answer each question correctly, then the total number of common question groups can be at most 225. Therefore, the minimum value sought is no less than 6.

If a person $A$ answers at most 3 questions correctly, then the remaining 20 people must answer at least 1 of these 3 questions correctly. By the pigeonhole principle, the question with the most correct answers must have at least 8 people answering it correctly.

If each person answers at least 4 questions correctly, then the 21 people answer at least 84 questions correctly. If no more than 6 people answer each question correctly, then 15 questions result in 90 correct answers. Since $90=21 \times 4+6$, the number of people who answer 4 questions correctly is at least 15, and the number of people who answer at least 5 questions correctly is at most 6.

Suppose $A$ answers 4 questions correctly, which are $a_{1}, a_{2}, a_{3}, a_{4}$. According to the given, the remaining 20 people each answer at least one of these 4 questions correctly. Since no more than 6 people answer each question correctly, the remaining 20 people can be divided into 4 groups, with the 5 people in the $i$-th group answering $a_{i}$ correctly but not the other 3 questions, $i=1,2,3,4$. This way, $a_{1}$, $a_{2}$, $a_{3}$, and $a_{4}$ are each answered correctly by exactly 6 people.

Since at most 6 people answer at least 5 questions correctly, by the pigeonhole principle, there is always one of the 4 groups where at most 1 person answers at least 5 questions correctly. Together with $A$, at least 5 people each answer 4 questions correctly. From the previous derivation, these 5 people, besides the 1 question they all answer correctly, each answer 3 other questions, totaling 15 questions that are all different. This leads to at least 16 different questions, which is impossible. Therefore, the minimum value sought is at least 7.
The table below lists the 21 people's correct answers:
\begin{tabular}{lll}
$(1,2,3,4)$ & $(3,5,9,13)$ & $(2,7,10,13)$ \\
$(1,5,6,7)$ & $(3,6,10,11)$ & $(3,5,9,13)$ \\
$(1,8,9,10)$ & $(3,7,8,12)$ & $(3,6,10,11)$ \\
$(1,11,12,13)$ & $(4,5,10,12)$ & $(3,7,8,12)$ \\
$(2,5,8,11)$ & $(4,6,8,13)$ & $(4,5,10,12)$ \\
$(2,6,9,12)$ & $(4,7,9,11)$ & $(4,6,8,13)$ \\
$(2,7,10,13)$ & $(2,6,9,12)$ & $(4,7,9,11)$
\end{tabular}

It is easy to see that every two people have at least 1 question in common and no more than 7 people answer each question correctly.
In summary, the question with the most correct answers has at least 7 people answering it correctly.