## Subject III. (30 points)

a) A three-digit number is reduced by 6 times if the middle digit is removed. Find the number.

b) Find all pairs of non-zero natural numbers whose sum is 111 and for which 111 is divisible by the difference between the two numbers.

Prof. Vasile Şerdean, Gherla Middle School No. 1
## Subject III.

a) $\overline{a b c}=6 \cdot \overline{a c}$. Then $100 a+10 b+c=60 a+6 c$

$$
\begin{aligned}
& 40 a+10 b=5 c \\
& 8 a+2 b=c \Rightarrow a=1 \Rightarrow 8+2 b=c \Rightarrow b=0, c=8
\end{aligned}
$$

b) Let $a, b \in N^{*}, a>b$ be the numbers sought. Then $a+b=111, (a-b) \in D_{111}=\{1,3,37,111\}$ (10p) Therefore $(a, b) \in\{(56,55) ;(57,54) ;(74,37)\}$