The repeating decimals <img alt="$0.abab\overline{ab}$" height="16" src="https://latex.artofproblemsolving.com/2/b/f/2bfde1c24303688d05fde20045cf565f340f6058.png" width="66" /> and <img alt="$0.abcabc\overline{abc}$" height="16" src="https://latex.artofproblemsolving.com/f/4/c/f4cb76705cb2df3b0d167c3d11b4421ff073e3ad.png" width="90" /> satisfy <img alt="\[0.abab\overline{ab}+0.abcabc\overline{abc}=\frac{33}{37},\]" height="37" src="https://latex.artofproblemsolving.com/1/a/b/1ab5008009f306f1d32c9fd9123a272d593dfcbd.png" width="228" /> where <img alt="$a$" height="10" src="https://latex.artofproblemsolving.com/c/7/d/c7d457e388298246adb06c587bccd419ea67f7e8.png" width="11" />, <img alt="$b$" height="14" src="https://latex.artofproblemsolving.com/8/1/3/8136a7ef6a03334a7246df9097e5bcc31ba33fd2.png" width="10" />, and <img alt="$c$" height="8" src="https://latex.artofproblemsolving.com/3/3/7/3372c1cb6d68cf97c2d231acc0b47b95a9ed04cc.png" width="8" /> are (not necessarily distinct) digits. Find the three digit number <img alt="$abc$" height="12" src="https://latex.artofproblemsolving.com/9/6/a/96a81938ca5e71faa6894d9d52ac776c9efbe75d.png" width="25" />.

Question No: 388

The repeating decimals $0.abab\overline{ab}$ and $0.abcabc\overline{abc}$ satisfy

$0.abab\overline{ab}+0.abcabc\overline{abc}=\frac{33}{37},$

where $a$ , $b$ , and $c$ are (not necessarily distinct) digits. Find the three digit number $abc$ .

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Tags: Algebra

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