4.36 In a competition with $n(\geqslant 2)$ participants lasting $k$ days, each participant's score each day is exactly $1,2, \cdots, n$ (no two participants have the same score), and at the end of $k$ days, each participant's total score is 26. Find all possible values of $(n, k)$.
(22nd Canadian Mathematical Competition, 1990)
[Solution] To calculate the total score of all participants, we have
$$
k(1+2+\cdots+n)=26 n .
$$

That is,
$$
k(n+1)=52 .
$$

Performing prime factorization on 52, we get $52=2^{2} \cdot 13$, which has 6 different divisors:
$$
1,2,4, \quad 13,26,52 \text {. }
$$

Since $n+1 \geqslant 3, k \geqslant 2$. (If $k=1$, i.e., only one day of competition, it is impossible for all participants to have the same score), the equation (1) has only three solutions:
$$
(n, k)=(25,2),(12,4),(3,13) .
$$

In fact, all three solutions are possible. We can list the daily scores of each participant for these three solutions.
$$
\begin{array}{lllll}
n=25, & k=2 \text { when, } & & \\
1, \quad 2, & 3, & \cdots, & 25, \\
25, & 24, & 23, & \cdots, & 1 . \\
n=12, & k=4 \text { when, } & & \\
1, \quad 2, & 3, & \cdots, & 12 \\
12, \quad 11, & 10, & \cdots, & 1 \\
1, \quad 2, & 3, & \cdots, & 12 \\
12, & 11, & 10, & \cdots, & 1
\end{array}
$$
(Day 2)
(Day 1)
(Day 2)
(Day 3)
(Day 4)
$$
n=3, \quad k=13 \text { when, }
$$
$1,2,3$
$1,2,3$
$1,2,3$
$1,2,3$
$2,3,1$
$2,3,1$
$2,3,1$
$2,3,1$
$2,1,3$
$3,1,2$
$3,1,2$
$3,1,2$
(Day 1)
(Day 2)
(Day 3)
(Day 4)
(Day 5)
(Day 6)
(Day 7)
(Day 8)
(Day 9)
(Day 10)
(Day 11)
$3,2,1$
(Day 12)
(Day 13)