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Instead of the three dots, write a digit to make the fraction reducible. (Find all possible cases.) 6...5/1...5 Answer ASAP please

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closed as off-topic by Lord Shark the Unknown, Alexander Gruber Apr 29 at 23:07

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If $n$ is the integer quotient and $x,y$ are the mising digits, we have $\frac{605+10x}{105+10y}=n$, or $$\tag1 605-105n=10\cdot (ny-x).$$ The right hand side is a multiple of $10$. For the left hand side to be a multiple of $10$, we need than $n$ is odd.

We have $\frac{6\ldots 5}{1\ldots 5}\le \frac{695}{105}<7$ and $\frac{6\ldots 5}{1\ldots 5}\ge \frac{605}{195}>3$, hence $3<n<7$, so by the previous result, $n=5$. Now $(1)$ becomes $ 80 = 10\cdot (5y-x)$, or $$ 5y=8+x$$ with the possibilities $x=2$, $y=2$ or $x=7$, $y=3$.

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