Irrational number multiplied by its fractional part becomes rational (SOLVED) Here's a Korean middle school midterm problem I've been struggling for quite some time now.
"$X$ is an irrational number such that $X>0$, and $Y$ is fractional part of $X$. If $$X^2+Y^2=27$$, find $X$."
So if we say
$X=(Y+N)$, such that $N=|X|$, we get
$(Y+N)^2+Y^2=27$; which is
$(Y^2+2YN+N^2)+Y^2$
$=2(Y^2)+2NY+N^2=27$.
As both $N^2$ and $27$ are rational, we must prove that there exists $Y$ and $N$ such that
$2(Y^2)+2NY$ is rational, which also means
$Y^2+NY$
$=Y(N+Y)$
$=XY$  is rational.
I can't find such irrational number which becomes a rational number when multiplied with its fractional part.
Can anyone tell how to solve this problem, or at least prove whether this is solvable/unsolvable?

$X$ is between 5 and 6 because $Y^2$ is between 0 and 1. Write $X = 5 + Y$ and you get a quadratic in $Y$ which you should be able to solve.
