Define relation R on Z+ by a R b, if r (a) = x(b), where x (a) = the number of positive (integer) divisors of a. For example, 2 R 3 and 4 R 25 but 5R9.
a) Verify that 2ft is an equivalence relation on Z+.
b) For the equivalence classes [a] and [b] induced by R, define operations of addition and multiplication by [a] + [b] = [a + b] and [a][b] = [ab]. Are these operations well- defined [that is, does a R c, b R d =>> (a + b) R (c + d), (ab) R (cd)]?