HW #2. Due  Wed, Feb 4, 1998.   Given on Fri, Jan 29.
Math 365. Number Theory.  Spring 1998.

Problem 1. Solve      156x=183  (mod  201).

Problem 2. Solve      36x=24  (mod  64).

Problem 3. Solve the equation   12x+17y=400
   in  positive  integers   (x,y).

Problem 4. Find all integers   x, 1<x<3000000,
   such that
     x = 1 (mod 101),  x = 2 (mod 102),
     x = 3 (mod 103).
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New Challenge problem.
   For  positive integers  a,b,  we denote by   S(a,b)
the set  of all possible sums  of the numbers  a  and  b,
with the arbitrary multiplicities.  For example,

  S(4,7)={4,7,8,11,12,14,15,16,18,19,20,21,22,23,24,...}.

Assume that  a,b  are two positive integers such
   that   (a,b)=1.  Let   c = ab-a-b.

Problem 1A.  Prove that   c   is not in the set   S(a,b).
Problem 1B.  Prove that  every  integer  larger than   c
   belongs  to  S(a,b).         (3 pts.)
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