2006 Seniors Pancyprian/2nd grade/Problem 2

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Problem

Find all three digit numbers $\overline{xyz}$($=100x+10y+z$) for which $\frac {7}{4}(\overline{xyz})=\overline{zyx}$.

Solution

$175x+\frac{35}{2}y+\frac{7}{4}z=100z+10y+x$

$174x+\frac{15}{2}y-\frac{393}{4}z=0$

$232x+10y-131z=0$

We can now make some conclusions. z is even, so z can be only 2, 4, 6, or 8, since 0 is an extraneous solution. Now if $z=2$, then $x=1$ and $y=3$. If $z=4$, then $x=2$ and $y=6$. If $z=6$, then $x=3$ and $y=9$. If $z=8$, then $x=4$, but there are no solutions for $y$. Thus only $123$, $246$, and $369$ work.

See also