Problem 2. During math class, each of the 25 students in the 5th grade receives a card on which a non-zero natural number is written. Each student divides the number on the card by 24 and communicates the remainder obtained to the teacher. The sum of the remainders obtained is 288. Student Daniel notices that the remainders obtained by his classmates are all different, and the quotient and remainder obtained by him are equal.

a) What number is written on Daniel's card?

b) Find the sum of the numbers written on the 25 cards, knowing that each student, except for Daniel, obtained a quotient that is 1 greater than the remainder.
Solution. a) Daniel's 24 colleagues obtained the 24 different possible remainders when divided by 24, namely $0,1,2, \ldots, 23$............................................1p

It follows that the remainder obtained by Daniel is $288-(0+1+2+3+\ldots+23)=12 \ldots$ 1p

Therefore, the number written on Daniel's cardboard is $24 \cdot 12+12=300 \ldots$ 1p

b) If one of Daniel's colleagues obtained the remainder $r, 0 \leq r \leq 23$, when divided by 24, then the number written on their cardboard is $24 \cdot(r+1)+r=25 r+24 \ldots \ldots \ldots$. . . . 1p

From this, we obtain that the sum of the numbers on the colleagues' cardboard is:

$$
\begin{aligned}
S & =(25 \cdot 0+24)+(25 \cdot 1+24)+(25 \cdot 2+24)+\ldots+(25 \cdot 23+24)= \\
& =25 \cdot(1+2+\ldots+23)+24 \cdot 24=7476
\end{aligned}
$$