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
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