For how many natural numbers x, is the expression: (x ² + 2x + 3) divisible by 35 ?

Let us substitute x+1 = y

By conditions of the problem:

(x^2 + 2x + 3)(Mod 35) = 0

Or, (y^2 + 2)(Mod 35) = 0

Or, y^2(Mod 35) = -2.....(i)

Now we observe that 7*5=35, where 7 and 5 are relatively coprime. Accordingly from (i), we obtain:

y^2(Mod 7) = 5....(ii), and:

y^2 (Mod 5) = 3....(iii)

However, we know that since 0, 1, 2, 4 correspond to the possible quadratic residues in the Mod 7 system, it follows that 5 is not a quadratic residue in that system. By way of similar arguments, it is evident that 3 is not a quadratic residue in the Mod 5 system. This leads to a contradiction.

Consequently, there do not exist any natural number x for which the expression: (x^2 + 2x + 3) is divisible by 35

*Edited on ***November 29, 2007, 12:21 pm**