# Instead of the three dots, write a digit to make the fraction reducible. (Find all possible cases.) 6…5/1…5 [closed]

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

## 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 $$3, so by the previous result, $$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$$.