B4. At a mathematics competition, 62 chocolates, 310 candies, 155 chewing gums, 93 oranges, and 186 juices were distributed among the participants so that each received an equal number of chocolates, candies, chewing gums, oranges, and juices.

a) How many participants were at the competition?

b) How many of each treat did each participant receive?

## Solutions to Problems and Scoring Guide

A participant who arrives at the solution through any correct method (even if the scoring guide does not anticipate it) receives all possible points.

A correct method is one that

- sensibly takes into account the wording of the problem,
- leads to the solution of the problem,
- is mathematically correct and complete.

A participant who has only partially solved the problem, from otherwise correct steps in solving but does not show the path to the final solution of the problem, cannot receive more than half of the possible points.

## PART I

The correct answers are recorded in the table. A correct answer by the participant is scored with 2 points, an incorrect answer with -1 point, and a blank field in the table with 0 points.

| Problem | A1 | A2 | A3 | A4 | A5 | A6 | A7 | A8 | A9 | A10 |
| :--- | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| Answer | $\mathrm{E}$ | $\mathrm{D}$ | $\mathrm{E}$ | $\mathrm{B}$ | $\mathrm{A}$ | $\mathrm{A}$ | $\mathrm{D}$ | $\mathrm{C}$ | $\mathrm{D}$ | $\mathrm{D}$ |
B4. The number of participants in the competition is determined by finding the common divisor of the numbers 62, 310, 155, 93, and 186. Since the prime factorizations of these numbers are \(62 = 2 \cdot 31\), \(310 = 2 \cdot 5 \cdot 31\), \(155 = 5 \cdot 31\), \(93 = 3 \cdot 31\), and \(186 = 2 \cdot 3 \cdot 31\), the greatest common divisor of all the listed numbers is 31, which is the number of participants in the mathematics competition.

Given the calculated number of participants in the mathematics competition, we can determine with simple calculations how many of each treat each participant received. Each participant received: \(62 \div 31 = 2\) chocolates, \(310 \div 31 = 10\) candies, \(155 \div 31 = 5\) chewing gums, \(93 \div 3 = 3\) oranges, and \(186 \div 31 = 6\) juices.

## Scoring: Total: 5 points

![](https://cdn.mathpix.com/cropped/2024_06_07_a1c9447e499e971c849bg-7.jpg?height=66&width=1594&top_left_y=1278&top_left_x=311)

Written answer, e.g.: There were 31 participants in the competition. ....................... 1 p

b) Written answer, e.g.: Each participant received: 2 chocolates, 10 candies, 5 chewing gums, 3 oranges, 6 juices.