perplexus.info :: Just Math : Always Divisible

Always Divisible (Posted on 2007-05-20)

Prove that for every integer x, there is an integer y such that (y^2-2)/(x^4+1) is an integer.

Submitted by Brian Smith

Rating: 4.0000 (1 votes)

Solution: (Hide)

Let y = x^3 - x. Then the expression can be written as:
((x^3-x)^2-2)/(x^4+1)
= (x^6 - 2x^4 + x^2 - 2)/(x^4+1)
= (x^2 - 2)*(x^4+1)/(x^4+1)
= x^2 - 2
Since x is an integer, x^2-2 is also an integer, therefore y=x^3-x will make the fraction an integer for all integer x.

Comments: ( You must be logged in to post comments.)

Subject	Author	Date
Puzzle Solution	K Sengupta	2007-05-21 05:22:34
Solution	Joel	2007-05-21 02:58:29
Only an observation	Dej Mar	2007-05-20 17:33:00

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