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Rational Rigor II (Posted on 2013-03-16) Difficulty: 3 of 5
Determine all possible rational numbers r such that 3r3 +10r2 - 3r is an integer.

No Solution Yet Submitted by K Sengupta    
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Solution allof tem - spoiler | Comment 1 of 3
Clearly, all integer  values of r satisfy the polinom being integer.
If we are lookin for fraction , it nust be n/3, n being an integer.

Then the polynom becomes  1/9 *n*(n^2+10*n-27).

n^2+10*n =n*(n+10), divisible by 9 iff n=0 mod 9 or 8 mod 9.

Therefore  r=  any integer

or r=   8/3,17/3.26/3    ...

and -1/3, -10/3, -19/3... etc.

Edited on March 17, 2013, 6:24 am
  Posted by Ady TZIDON on 2013-03-17 05:06:00

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