$m$ and $n$ range over all the integers — positive, negative or zero. What is the smallest POSITIVE value that $21m+35n$ can take?

Every value of $21m+35n$ is a multiple of 7, because 7 divides both 21 and 35, so nothing smaller than 7 is reachable. And 7 is actually reached: $21\times 2+35\times (-1)=42-35=7$. In general the smallest positive value of am + bn is exactly HCF(a, b), which here is HCF(21, 35) = 7. Answering 1 quietly assumes 21 and 35 are co-prime, which they are not, and 56 is just 21 + 35 with $m=n=1$.