[Number Theory]
Problem: Compute the smallest positive integer $N$ such that $20 N$ is a multiple of 14 and $14 N$ is a multiple of 20 .

Solution: Because $\operatorname{gcd}(14,20)=2$, the problem is equivalent to computing the smallest positive integer $N$ such that $7 \mid 10 N$ and $10 \mid 7 N$. Thus $7 \mid N$ and $10 \mid N$, and the desired value of $N$ is $\operatorname{lcm}(7,10)=\mathbf{7 0}$.

Final answer: 70