Difference between revisions of "2006 Seniors Pancyprian/2nd grade/Problem 2"
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== Solution == | == Solution == | ||
− | {{ | + | <math>175x+\frac{35}{2}y+\frac{7}{4}z=100z+10y+x</math> |
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+ | <math>174x+\frac{15}{2}y-\frac{393}{4}z=0</math> | ||
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+ | <math>232x+10y-131z=0</math> | ||
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+ | 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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== See also == | == See also == | ||
Revision as of 10:46, 12 August 2008
Problem
Find all three digit numbers () for which .
Solution
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 , then and . If , then and . If , then and . If , then , but there are no solutions for . Thus only , , and work.