## SUBJECT III

Determine the numbers formed of two or three digits which, when divided by 13, give the quotient $a$ and the remainder $b$, and when divided by 11, give the quotient $b$ and the remainder $a$.
## Solution:

Let $A$ be the number we are looking for, then: $A=13a+b, b \leq 12$ $.1 \text{p}$

and $A=11b+a, a \leq 10$.......................................................................................... $1 \text{p}$

From which $13a+b=11b+a \Rightarrow 6a=5b$, ................................................................... $2 \text{p}$

![](https://cdn.mathpix.com/cropped/2024_06_07_cc8ac07229c0dd5d256fg-3.jpg?height=74&width=1373&top_left_y=1162&top_left_x=427)

The numbers we are looking for are: 71 and 142. .............................................................................. $1 \text{p}$