## SUBJECT II

a) Dividing the natural number a by the natural number b results in a quotient of 14 and a remainder of 18. If the difference between the numbers a and a-3b is equal to 135, show that the number 2a is a perfect square.

Gazeta matematica

b) How many three-digit numbers, when divided by a natural number b, give a quotient of 14 and a remainder of 18?

Calculate their sum.
## SUBJECT II

$a=b \cdot 14+18$

$a-3 b=b \cdot 11+18$

$a-(a-3 b)=b \cdot 14+18-(b \cdot 11+18)$

$$
\begin{aligned}
& =3 b=135 \\
& =>b=45 \quad 1 \text { point }
\end{aligned}
$$

$a=b \cdot 14+18$

$$
\begin{array}{rlr}
=45 \cdot 14+18=648=2^{3} \cdot 3^{4} & \} & 1 \text { point } \\
2 a=2^{2} \cdot 3^{4}=\left(2^{2}\right)^{2} \cdot\left(3^{2}\right)^{2}=\text { p.p } & \} & 1 \text { point }
\end{array}
$$