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I would like to find some positive integer solutions to an equation in the form $Ax + By = C.$ I have already seen some methods for doing this, such as the one outlined in this Math.SE post.

What I am interested in is the method shown here. The first example is the one I was looking at. I can follow it up to fig [1.2]. I don't understand what the author means by "Reducing the right-hand side to integers and fractions." I would greatly appreciate some help in understanding this process. Also, is there a name that I can Google for the method used in this article? Thanks.

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It's called a mixed fraction. $$\dfrac{4238 - 97y}{95} = 44 \ \ \dfrac{58}{95} - (1 \ \ \dfrac{2}{95})y$$ –  The Chaz 2.0 Apr 19 '12 at 0:10
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up vote 1 down vote accepted

It's just a funny way to call a pretty elementary arithmetic operation: $$\frac{4238-97y}{95}=\frac{44\cdot 95+58-(95y)-2y}{95}=\frac{44\cdot 95}{95}-\frac{95y}{95}+\frac{58-2y}{95}=$$ $$=44-y+\frac{58-2y}{95}$$

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This is integer division with remainder en.wikipedia.org/w/index.php?title=Division_with_remainder. Now for integer solutions you need $2y \equiv 58 \pmod {95}$ or $y=29+95t$ for some integer $t$ as is shown on the page OP linked to. –  Ross Millikan Jul 4 '12 at 3:21
    
Yup, yet the OP just asked about the process the page he linked to was carrying out. Apparently, and hopefully, he can follow the rest of the argument –  DonAntonio Jul 4 '12 at 3:26
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