soln: claim 1: If x belongs to set A, nx belongs to a where n is any integer: proof: x belongs to A - > x-x belongs to A -> 0 belongs to A-> 0-x belongs to A-> -x belongs to A now we can subtract -x as many times as we like from x to get all positive x and subtract x to get all negative multiples of x. claim 2: If 1 can be expressed as the product of 2 values, all integers can be expressed as product of 2 integers proof: if k and l belong to A such that kl = 1, for any integer N, we can select Nk (from claim 1) and l and Nkl = kl * N = 1*N = N claim 3:1 can be expressed as product of 2 elements. proof: we know that root 2 is part of A (root(1^2 + 1)) hence from claim 1 3root(2) belongs to set A which can be expressed as root(18) similarily root(17) belongs to A (root(4^2 + 1) hence we get -root(17) also belongs to A now this implies (root(18) - root(17)) as well as (root(18) + root(17)) belong to A product of these 2 elements is 1 (using (a-b)(a+b) = a^2 - b^2) Hence proved.