## SUBJECT 4

A natural number with four digits has its first two digits identical, and the units digit is 5. This number is divided by a two-digit number, and the remainder is 98. Calculate the dividend, divisor, and quotient.

## Note:

Working time: 2 hours.

All subjects are mandatory.

Each subject is graded from 0 to 7.

No points are given ex officio.

## MATHEMATICS OLYMPIAD

Local stage - Constanța, 15.02.2015

5th Grade

## Grading and marking criteria
## Subject 4.

![](https://cdn.mathpix.com/cropped/2024_06_07_514927207678d9ecd2fbg-2.jpg?height=72&width=1563&top_left_y=2117&top_left_x=246)
Since $x$ has two digits, it follows that $x=99$, so $\overline{a a b 5}=99 \cdot q+98$ ..... $2p$

It follows that the last digit of $99 \cdot q+98$ is 5, so the last digit of $q$ is 3, but $1105 \leq \overline{a a b 5} \leq 9995 \Leftrightarrow 1105 \leq 99 \cdot q+98 \leq 9995$, from which it is obtained that $11 \leq q \leq 99$ and, since the last digit of $q$ is 3, it follows that $q \in\{13,23,33, \ldots, 93\}$

2p
Therefore, the dividend is equal to 3365, the divisor is equal to 99, and the quotient is equal to 33. ..... $2p$