# Problem 5.

An electric cable 21 meters long is cut into 21 pieces. For any two pieces, their lengths differ by no more than a factor of three. What is the smallest $m$ such that there will definitely be two pieces whose lengths differ by no more than a factor of $m$?
# Solution.

If among the pieces of cable there are at least two that differ in length, then by taking the ratio of the smaller to the larger, we get that $m \leq 1$.

However, the cable can be cut in such a way that all pieces are equal. This implies that if $m<1$, then a way of cutting has been found where no two pieces differ by more than a factor of $m$.

The constructed example proves that the smallest value of $m$ is 1.

Answer: $m=1$.