Answer Happy

For some of the answers below, you will need to enter a set. To enter the set {1, 2, 3}, use either the Maple syntax {1, 2, 3} or the Numbas syntax set (1, 2, 3). (a) Matthew is trying to solve a modular congruence that Dr Aritz has written up on the blackboard: 72x92 (mod 332). First, Matthew writes the solution as an integer x with respect to a smallest possible modulus k: x = (mod k). Next, Matthew writes the solution as a set of integers {x1,x2,...} with respect to the original modulus 332 : x E (mod 332). (b) Matthew looks up at the board to copy down the next question, but Dr Aritz has already started cleaning the board! Matthew copies down what they can, putting question marks (?) where they were not able to copy down certain numbers. The series of question marks (???) could represent a list of zero, one, or more numbers: 48x = (mod 153) has the solution x = {69, ???} (mod 153). Help Matthew find the complete solution set {x₁,x2,...} with respect to the modulus 153: XE (mod 153). (c) Dr Aritz turns to his lecture notes for the next example, but realises he has accidentally spilled homebrand Coke on his notes and cannot read all the numbers. The example looks like this, where again a question mark (?) represents an unknown number, and a series of question marks (???) could represent a list of zero, one, or more numbers: 28x = ? (mod ?) has the solution x = { ??? |} (mod ?), so there are n = ? different solutions in the original modulus. What are the possible values for the size of the solution set? That is, taking n as the number of solutions for x in the original modulus, write the set of all possible values for n as a set of integers {n₁, n2,...} : ne