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Divisibility (Posted on 2004-01-06) Difficulty: 3 of 5
For how many natural numbers x, is the expression: (x + 2x + 3) divisible by 35 ?

See The Solution Submitted by Ravi Raja    
Rating: 2.5000 (6 votes)

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Solution Alternative Methodology | Comment 9 of 11 |

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
  Posted by K Sengupta on 2007-11-29 11:12:38

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