13.439 If the desired two-digit number is increased by 46, the resulting number will have a product of its digits equal to 6. Find this number given that the sum of its digits is 14.
Solution. Let $\overline{x y}$ be the desired number $z$ - the number obtained from $\overline{x y}$ by adding 46. Then $y=14-x, 10 x+y+46=z \Rightarrow 9 x+60=z$. Consider two cases.

1) $z=\overline{a b}$. Then $9 x+60=10 a+b=9 a+(a+b)$, i.e., $a+b$ is divisible by 3. But by the condition $a \cdot b=6$. Hence, one of the numbers $a, b$ is divisible by 3, and therefore, the other is also divisible by 3, which is impossible.
2) $z=\overline{c a b}$. Obviously, $c=1$. Hence, $9 x+60=100+10 a+b \Rightarrow$

$\Rightarrow 9 x=40+10 a+b \Rightarrow 9(x-a-4)=4+a+b$, i.e., $a+b+4$ is divisible by 9.

But $a \leq 6, b \leq 6 \Rightarrow a+b+4 \leq 16 \Rightarrow a+b+4=9 \Rightarrow a+b=5$.

Taking into account $a \cdot b=6 \Rightarrow\left[\begin{array}{l}a=2, b=3, \\ a=3, b=2\end{array} \Rightarrow\left[\begin{array}{l}z=123, \\ z=132\end{array} \Rightarrow\left[\begin{array}{l}\overline{x y}=77, \\ \overline{x y}=86 .\end{array}\right.\right.\right.$

Answer: 77 or 86.