3. While solving the task, Josip had to multiply two two-digit numbers. However, he made a mistake by writing the digits of one number in reverse order. He obtained a product that was 3816 greater than the correct result of the task. What is the correct result?
First method:

Let $\overline{a b}$ and $\overline{c d}$ be the given two-digit numbers. According to the conditions of the problem, we have:

$\overline{a b} \cdot \overline{c d} + 3816 = \overline{a b} \cdot \overline{d c}$, or $(10 a + b) \cdot (10 c + d) + 3816 = (10 a + b) \cdot (10 d + c)$.

Further, we have:

$100 a c + 10 a d + 10 b c + b d + 3816 = 100 a d + 10 a c + 10 b d + b c$,

$90 a c + 9 b c + 3816 = 90 a d + 9 b d$,

$10 a c + b c + 424 = 10 a d + b d$,

$c(10 a + b) + 424 = d(10 a + b)$,

$c \cdot \overline{a b} + 424 = \overline{a b} \cdot d$,

$\overline{a b}(d - c) = 424$

This can be solved in two ways:
a) $c$ and $d$ are digits, so their difference is certainly less than 9 (it cannot be 9 because then $c$ would be 0, which is not possible) and different from zero (due to the product which has a value of 424).
I. $d - c = 1 \rightarrow \overline{a b} = 424$, which is not possible because $\overline{a b}$ is a two-digit number

II. $d - c = 2 \rightarrow \overline{a b} = 212$, which is not possible because $\overline{a b}$ is a two-digit number

III. $d - c = 3 \rightarrow$ not possible because 424 is not divisible by 3

IV. $d - c = 4 \rightarrow \overline{a b} = 106$, which is not possible because $\overline{a b}$ is a two-digit number
V. $d - c = 5 \rightarrow$ not possible because 424 is not divisible by 5

VI. $d - c = 6 \rightarrow$ not possible because 424 is not divisible by 6

VII. $d - c = 7 \rightarrow$ not possible because 424 is not divisible by 7

VIII. $d - c = 8 \rightarrow \overline{a b} = 53$

From $\overline{a b} = 53$ and $d = 8 + c$ we get $c = 1$ and $d = 9$, i.e., $\overline{c d} = 19$. The correct product is $53 \cdot 19 = 1007$.

b) By factoring the number 424 into prime factors, we get: $424 = 2 \cdot 2 \cdot 2 \cdot 53$. From this, the two-digit number $\overline{a b} = 53$, the difference of the digits is $d - c = 8$, so $c = 1, d = 9$, i.e., $\overline{c d} = 19$.

The correct product is $53 \cdot 19 = 1007$.