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Difference between revisions of "2018 AIME I Problems/Problem 2"

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The number <math>n</math> can be written in base <math>14</math> as <math>\underline{a}\text{ }\underline{b}\text{ }\underline{c}</math>, can be written in base <math>15</math> as <math>\underline{a}\text{ }\underline{c}\text{ }\underline{b}</math>, and can be written in base <math>6</math> as <math>\underline{a}\text{ }\underline{c}\text{ }\underline{a}\text{ }\underline{c}\text{ }</math>, where <math>a > 0</math>. Find the base-<math>10</math> representation of <math>n</math>.
  
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==Solutions==
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==Solution Algebra==
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We have these equations:
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<math>196a+14b+c=225a+15c+b=222a+37c</math>.
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Taking the last two we get <math>3a+b=22c</math>. Because <math>c \neq 0</math> otherwise <math>a \ngtr 0</math>, and <math>a \leq 5</math>, <math>c=1</math>.
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Then we know <math>3a+b=22</math>.
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Taking the first two equations we see that <math>29a+14=13b</math>. Combining the two gives <math>a=4, b=10</math>. Then we see that <math>222*4+37*1=\boxed{925}</math>.
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-expiLnCalc

Revision as of 18:31, 7 March 2018

The number $n$ can be written in base $14$ as $\underline{a}\text{ }\underline{b}\text{ }\underline{c}$, can be written in base $15$ as $\underline{a}\text{ }\underline{c}\text{ }\underline{b}$, and can be written in base $6$ as $\underline{a}\text{ }\underline{c}\text{ }\underline{a}\text{ }\underline{c}\text{ }$, where $a > 0$. Find the base-$10$ representation of $n$.

Solutions

Solution Algebra

We have these equations: $196a+14b+c=225a+15c+b=222a+37c$. Taking the last two we get $3a+b=22c$. Because $c \neq 0$ otherwise $a \ngtr 0$, and $a \leq 5$, $c=1$.

Then we know $3a+b=22$. Taking the first two equations we see that $29a+14=13b$. Combining the two gives $a=4, b=10$. Then we see that $222*4+37*1=\boxed{925}$.

-expiLnCalc

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