Consider the following C code. Assume that unsigned long int type length is 64 bits.

unsigned long int fun(unsigned long int n){
     unsigned long int i, j = 0, sum = 0;
     for (i = n; i > 1; i = i/2) j++;
     for ( ; j > 1; j = j/2) sum++;
     return(sum);
}

The value returned when we call fun with the input $${2^{40}}$$ is

The set of all recursively enumerable languages is

Let $$N$$ be an $$NFA$$ with $$n$$ states. Let $$k$$ be the number of states of a minimal $$DFA$$ which is equivalent to $$N.$$ Which one of the following is necessarily true?

Consider the following languages:

$$\,\,\,\,\,\,\,\,{\rm I}.\,\,\,\,\,\,\,$$ $$\left\{ {{a^m}{b^n}{c^p}{d^q}} \right.|m + p = n + q,$$ where $$\left. {m,n,p,q \ge 0} \right\}$$
$$\,\,\,\,\,\,{\rm II}.\,\,\,\,\,\,\,$$ $$\left\{ {{a^m}{b^n}{c^p}{d^q}} \right.|m = n$$ and $$p=q,$$ where $$\left. {m,n,p,q \ge 0} \right\}$$
$$\,\,\,\,{\rm III}.\,\,\,\,\,\,\,$$ $$\left\{ {{a^m}{b^n}{c^p}{d^q}} \right.|m = n = p$$ and $$p \ne q,$$ where $$\left. {m,n,p,q \ge 0} \right\}$$
$$\,\,\,\,{\rm IV}.\,\,\,\,\,\,\,$$ $$\left\{ {{a^m}{b^n}{c^p}{d^q}} \right.|mn = p + q,$$ where $$\left. {m,n,p,q \ge 0} \right\}$$

Which of the languages above are context-free?