For all problems, you may assume the following functions are computable: basic functions; constant functions; x + y; xy; xy; x · y (cutoff); sg(x); s ̄g(x); |x − y|; min(x, y); max(x, y); rm(x, y) (remainder of y/x, where rm(0,y) = y); qt(x,y) (quotient of y/x, where qt(0,y) = 0); div(x,y) (“x divides y”, where 0|0 but 0 doesn't divide y if y > 0); D(x) (number of divisors of x); Pr(x) (“x is prime”); px (xth prime); (x)y. You may use Substitution, Theorem 3.2, Recursion, bounded sums, bounded products, bounded minimalization, and minimalization. You may also use “definition by cases” (but make sure you state and verify all necessary conditions).