5. Ari is watching several frogs jump straight towards him from the same place. Some frogs are larger, and some are smaller. When the larger frog makes 7 equal jumps, it is still $3 \mathrm{~cm}$ away from Ari. When the smaller frog makes 10 equal jumps, it is still $1 \mathrm{~cm}$ away from Ari. If the frogs are initially more than $2 \mathrm{~m}$ away from Ari, what is the smallest possible length of the jump of the smaller frogs? The length of each jump in centimeters is a natural number.

No pocket calculators or any reference materials are allowed.

## Ministry of Science and Education of the Republic of Croatia Agency for Education and Education Croatian Mathematical Society

## County Competition in Mathematics February 26, 2024.  5th grade - elementary school

Each task is worth 10 points. In addition to the final result, the procedure is also scored. To earn all points, it is necessary to find all solutions and confirm that there are no others, write down the procedure, and justify your conclusions.
First solution.

The holes are located at a distance from Ari that is more than $2 \mathrm{~m}=200 \mathrm{~cm}$.

Let's determine which numbers can be the distances of the smaller holes from Ari, and the distances of the larger holes from Ari.

The distance of the smaller holes from Ari is a number obtained by adding 1 to a multiple of 10 (i.e., a number that gives a remainder of 1 when divided by 10).

$1 \text{ POINT}$

Since 200 is a multiple of 10, the next multiples are $210, 220, 230, 240, 250, \ldots$

The distances of the larger holes from Ari can be the numbers: 201, 211, 221, 231, 241, 251, ... 2 POINTS

The distance of the larger holes from Ari is a number obtained by adding 3 to a multiple of 7 (i.e., a number that gives a remainder of 3 when divided by 7).

1 POINT

$200: 7=28$ with a remainder of 4

The first multiple of 7 greater than 200 is 203, and the next ones are $210, 217, 224, 231, 238, 245, \ldots$

The distances of the larger holes from Ari can be the numbers: 206, 213, 220, 227, 234, 241, 248, ...

2 POINTS

Since the holes are jumped from the same place, we are looking for the smallest number that can be both the distance of the smaller holes and the distance of the larger holes from Ari.

1 POINT

This number is 241, so the length of the jump of the smaller holes is

$(241-1): 10=24 \mathrm{~cm}$

TOTAL 10 POINTS