Which of the following claims are true and which are false. justify your answer by giving either a (short) proof or a counterexample. (a) in any maximum flow there are no directed cycles that carry positive flow. (a cycle (e1,.ek) carries positive flow iff f(et)> (ek) > 0.) (b) there always exists a maximum flow without cycles carrying positive flow. (c) if all edges in a graph have distinct capacities, there is a unique maximum flow. (d) in a directed graph with at most one edge between each pair of vertices, if we replace each directed edge by an undirected edge, the maximum flow remains unchanged. (e) if we multiply all edge capacities by a positive number a, the minimum cut remains unchan ged.

Write a program that uses a random number generator to generate a two digit positive integer and allows the user to perform one or more of the following operations: a. double the number. b. reverse the digits of the number. c. raise the number to the power of 2, 3, or 4. d. sum the digits of the number. e. if the number is a two-digit number, then raise the first digit to the power of the second digit. f. if the number is a three-digit number and the last digit is less than or equal to 4, then raise the first two digits to the power of the last digit. after performing an operation if the number is less than 10, add 10 to the number. also, after each operation determine if the number is prime. each successive operation should be performed on the number generated by the last operation. your program should not contain any global variables and each of these operations must be implemented by a separate function. also, your program should be menu driven. 7. (fraction calculator) write a program that