It is straightforward to solve for x any equation of the for

It is straightforward to solve for x any equation of the form x +n a = b in Zn and to see that the result will be a unique value of x. However, in the discussion of Exercise 2.1-6, we saw that 0, 3, 6, and 9 are all solutions to the equation 4 · 12 x = 0. Are there any integral values of a and b, with 1 less than or equal to a and b, which are both less than 12, for which the equation a · 12 x = b does not have any solutions in Z12? If there are, give one set of values for a and b. If there are not, explain how you know this. .

Solution

x=b/12a , hence for an integral value of x , b has to be multile of 12. To start with lowest value for a , i.e a =1 , b=12 , as we increase value of a as integers 1,2,3,.. the value of b increases ro 12,24,36.... Hence no integral solution exist for x for integal values of a,b which are both less than 12