<p>Find the three-digit positive integer <img alt="$\underline{a}\,\underline{b}\,\underline{c}$" height="15" src="https://latex.artofproblemsolving.com/6/8/0/68031d8e27e0ec70705d61b7621f7f5080bcf20f.png" width="31" /> whose representation in base nine is <img alt="$\underline{b}\,\underline{c}\,\underline{a}_{\,\text{nine}},$" height="19" src="https://latex.artofproblemsolving.com/2/a/e/2aeff01151702ab75db639d616979291f135a919.png" width="66" /> where <img alt="$a,$" height="11" src="https://latex.artofproblemsolving.com/7/c/8/7c8acfd7d5ee559262593701b8dbd02e43ad96e3.png" width="13" /> <img alt="$b,$" height="16" src="https://latex.artofproblemsolving.com/5/b/1/5b1d6265e67657b5886ce257671d45ff9c0282eb.png" width="12" /> and <img alt="$c$" height="8" src="https://latex.artofproblemsolving.com/3/3/7/3372c1cb6d68cf97c2d231acc0b47b95a9ed04cc.png" width="8" /> are (not necessarily distinct) digits.</p>

Question No: 430

Find the three-digit positive integer $\underline{a}\,\underline{b}\,\underline{c}$ whose representation in base nine is $\underline{b}\,\underline{c}\,\underline{a}_{\,\text{nine}},$ where $a,$ $b,$ and $c$ are (not necessarily distinct) digits.

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Tags: Number Theory

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