# 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>. | ||

+ | ==Solutions== | ||

+ | |||

+ | ==Solution Algebra== | ||

+ | |||

+ | We have these equations: | ||

+ | <math>196a+14b+c=225a+15c+b=222a+37c</math>. | ||

+ | 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>. | ||

+ | |||

+ | Then we know <math>3a+b=22</math>. | ||

+ | 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>. | ||

+ | |||

+ | -expiLnCalc |

## Revision as of 18:31, 7 March 2018

The number can be written in base as , can be written in base as , and can be written in base as , where . Find the base- representation of .

## Solutions

## Solution Algebra

We have these equations: . Taking the last two we get . Because otherwise , and , .

Then we know . Taking the first two equations we see that . Combining the two gives . Then we see that .

-expiLnCalc