2021 CIME I Problems/Problem 2

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Problem 2

For digits $a, b, c,$ with $a\neq 0,$ the positive integer $N$ can be written as $\underline{a}\underline{a}\underline{b}\underline{b}$ in base $9,$ and $\underline{a}\underline{a}\underline{b}\underline{b}\underline{c}$ in base $5$. Find the base-$10$ representation of $N$.

Solution

Consider the different representations of the number and equate them: \[(9^3 + 9^2) a +(9+1)b = (5^4+5^3) a + (5^2+5)b+c\] \[(810)a+10b = (750)a+30b+c\] \[60 a - 20b-c=0\]

Note that c can't contribute since it is less than 5 so $c=0$ Next note that $b = 3a$ since $b<5$ and $a>0$ the only solution is $b=3$,$a=1$ Thus in base 10 the number is $810+30=840$

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