Problem 2. Find the quotient and remainder when dividing the expression

$$
1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6 \cdot 7 \cdot 8 \cdot 9 \cdot 10 \cdot 11 \cdot 12+75
$$

by the number 35.
Solution. First method. The expression $1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6 \cdot 7 \cdot 8 \cdot 9 \cdot 10 \cdot 11 \cdot 12+75$ can be written in the following form:

$$
\begin{aligned}
& 1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6 \cdot 7 \cdot 8 \cdot 9 \cdot 10 \cdot 11 \cdot 12+35 \cdot 2+5= \\
&=35 \cdot 1 \cdot 2 \cdot 3 \cdot 4 \cdot 6 \cdot 8 \cdot 9 \cdot 10 \cdot 11 \cdot 12+35 \cdot 2+5 \\
&=35 \cdot(1 \cdot 2 \cdot 3 \cdot 4 \cdot 6 \cdot 8 \cdot 9 \cdot 10 \cdot 11 \cdot 12+2)+5
\end{aligned}
$$

According to this, the desired quotient is $1 \cdot 2 \cdot 3 \cdot 4 \cdot 6 \cdot 8 \cdot 9 \cdot 10 \cdot 11 \cdot 12+2=13685762$, and the remainder is 5.

Second method. We will calculate the sum

$$
1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6 \cdot 7 \cdot 8 \cdot 9 \cdot 10 \cdot 11 \cdot 12+75
$$

In this case, we get

$1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6 \cdot 7 \cdot 8 \cdot 9 \cdot 10 \cdot 11 \cdot 12+75=479001600+75=479001675$.

If we divide by 35, we get

$479001675: 35=13685762$

$\underline{35}$

129 $\qquad$
5

According to this

$$
479001675=13685762 \cdot 35+5
$$

Thus, the desired quotient is 13685762 and the remainder is 5.