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Indivisibility by 529 (Posted on 2016-07-25) Difficulty: 3 of 5
Prove that for any integer x, the value of x^2+7x+18 is not divisible by 529.

  Submitted by Brian Smith    
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Solution: (Hide)
First note that 529 is the square of the prime number 23.

Rearrange the quadratic x^2+7x+18 into (x+15)*(x-8) + 23*6. x+15 and x-8 are either both multiples of 23 or neither are. Then (x+15)*(x-8) is either a multiple of or coprime to 529.

23*6 is a multiple of 23 but not 529. If (x+15)*(x-8) is a multiple of 529 then the total is a multiple of 23 but not 529. Also if (x+15)*(x-8) is not a multiple of 529 then the total is coprime to 23 and 529. In either case x^2+7x+18 is not a multiple of 529.

Comments: ( You must be logged in to post comments.)
  Subject Author Date
Some Thoughtsre: completing the squareAdy TZIDON2016-07-25 23:34:57
completing the squarexdog2016-07-25 17:47:27
Brute force.broll2016-07-25 14:38:59
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