## Task 4 - 080624

Three friends are preparing for the "Kleine Friedensfahrt." They train on a circular track. Their start occurs at the same time and in the same direction at the starting line $S$. Manfred completed the first lap in exactly 3 minutes, Klaus in exactly $3 \frac{3}{4}$ minutes, and Helmut in exactly 5 minutes.

a) After how many minutes would the three friends first reach the starting line $S$ simultaneously again, assuming that Manfred takes exactly 3 minutes per lap, Klaus exactly $3 \frac{3}{4}$ minutes, and Helmut exactly 5 minutes for all subsequent laps?

b) How many laps would each of them have completed by then?
(a) After 15 minutes, the three friends would meet at the starting line $S$ for the first time under the conditions of the problem.

Proof: Helmut needed exactly 300 seconds for each lap, Klaus exactly 225 seconds, and Manfred exactly 180 seconds. The least common multiple (LCM) of 225, 300, and 180 is 900, and 900 seconds equal 15 minutes.

(b) In 15 minutes, Helmut would complete exactly 3 laps, Klaus exactly 4 laps, and Manfred exactly 5 laps.

Solutions of the II. Round 1968 taken from [5]

### 3.10 IX. Olympiad 1969

### 3.10.1 I. Round 1969, Class 6