Difference between revisions of "2006 Seniors Pancyprian/2nd grade/Problem 2"

Revision as of 09:46, 12 August 2008

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.

@@ Line 3: / Line 3: @@
 == Solution ==
-{{solution}}
+<math>175x+\frac{35}{2}y+\frac{7}{4}z=100z+10y+x</math>
+<math>174x+\frac{15}{2}y-\frac{393}{4}z=0</math>
+<math>232x+10y-131z=0</math>
+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 <math>z=2</math>, then <math>x=1</math> and <math>y=3</math>. If <math>z=4</math>, then <math>x=2</math> and <math>y=6</math>. If <math>z=6</math>, then <math>x=3</math> and <math>y=9</math>. If <math>z=8</math>, then <math>x=4</math>, but there are no solutions for <math>y</math>. Thus only <math>123</math>, <math>246</math>, and <math>369</math> work.
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Difference between revisions of "2006 Seniors Pancyprian/2nd grade/Problem 2"

Revision as of 09:46, 12 August 2008

Problem

Solution

See also