5. When the saleswoman was asked how many bunches of violets she had in her basket, she said: "If I take 2 bunches from the basket, 1 remains. If I take 3 or 4 bunches, 1 also remains. If I take 7 bunches, none remain. I am sure there are fewer than 100." How many bunches of violets did the saleswoman have in her basket?

## You have 120 minutes to solve the problems.

Solve each problem independently on the provided paper, with each problem on a separate page. Do not sign the sheets, write only your code. Write your answers in pen, clearly and neatly. Draw graphs with a pencil. If you solve a problem in multiple ways, clearly indicate which solution the evaluator should grade. If you make a mistake, strike it through.

Each problem's solution will be graded from 0 to 6 points.

The National Competition Committee wishes you great success.

## 7th Competition in Mathematics Knowledge for Students of Secondary Technical and Vocational Schools

National Competition, April 21, 2007

## Problems for 2nd Year
5. Let $x$ be the number of bunches of violets. Using the basic theorem of division, we write $x=2k+1=3s+1=4t+1=7n$. We are looking for the least common multiple of the numbers 2, 3, and 4. The desired multiple is 12. We add 1 to the multiples of 12. Possible values are $13, 25, 37, 49, 61, 73, 85$ and 97. We find that only 49 is a multiple of 7.

Write: the number of bunches is $x$..........................................................................................................

![](https://cdn.mathpix.com/cropped/2024_06_07_cd02ddf507985fa73630g-07.jpg?height=55&width=1642&top_left_y=2000&top_left_x=204)

![](https://cdn.mathpix.com/cropped/2024_06_07_cd02ddf507985fa73630g-07.jpg?height=57&width=1639&top_left_y=2053&top_left_x=206)

![](https://cdn.mathpix.com/cropped/2024_06_07_cd02ddf507985fa73630g-07.jpg?height=60&width=1639&top_left_y=2097&top_left_x=208)

![](https://cdn.mathpix.com/cropped/2024_06_07_cd02ddf507985fa73630g-07.jpg?height=57&width=1636&top_left_y=2150&top_left_x=210)

## Second Year