For how many natural numbers n from 1 to 100 is HCF(n, 36) equal to 4?

$n$ must be a multiple of 4, say $n=4m$ with $m$ ≤ 25. Since $36=4\times 9$ has only $2^2$, HCF(4$m$, 36) = 4 × HCF$(m$, 9). This equals 4 exactly when $m$ is not a multiple of 3. Of $m=1$ to 25, eight are multiples of 3, leaving 25 − 8 = 17.