## Task 1 - 090721

Father and son walk side by side. In the same time it takes the father to take 4 steps, the son takes 5 steps, and in this time, both cover exactly the same distance. The average stride length of the father is $80 \mathrm{~cm}$.

a) How long is the average stride length of the son?

b) Assuming both start with their right foot at the same time, after how many steps of the father do both step with their left foot at the same time for the first time?
a) With 4 steps, the father covers $320 \mathrm{~cm}$; because $4 \cdot 80 \mathrm{~cm}=320 \mathrm{~cm}$. Since the son needs 5 steps for the same distance, his average step length is $64 \mathrm{~cm}$ because $320: 5=64$.

b) The father and the son step simultaneously exactly when the father completes a (positive integer) multiple of 4 steps. This is because the son completes a whole number of steps at the same time. This happens exactly when both have completed an even number of steps, which means they step with the left foot. For the father, this is the case for every multiple of 4, and for the son, it is exactly for all even multiples of 5. The smallest (positive) even multiple of 5 is the double.

Therefore, the father and the son first step with the left foot simultaneously after the 8th step of the father.