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![\frac{5n+6}{8n+7}=\frac{5n}{8n+7} + \frac{6}{8n+7}= \ (*) \frac{5n+6}{8n+7}=\frac{5n}{8n+7} + \frac{6}{8n+7}= \ (*)](https://tex.z-dn.net/?f=%5Cfrac%7B5n%2B6%7D%7B8n%2B7%7D%3D%5Cfrac%7B5n%7D%7B8n%2B7%7D+%2B+%5Cfrac%7B6%7D%7B8n%2B7%7D%3D+%5C+%28%2A%29)
![\frac{5n}{8n+7} = A + \frac{B}{8n+7}; \ \ \ \ A \cdot (8n+7) + B=5n; \\ \\ \left \{ {{8A=5} \atop {7A+B=0}} \right. \ \ \left \{ {{A=\frac{5}{8}} \atop {B=-\frac{35}{8}}} \right. \\ \\ \frac{5}{8} - \frac{35}{8 \cdot (8n+7)} \frac{5n}{8n+7} = A + \frac{B}{8n+7}; \ \ \ \ A \cdot (8n+7) + B=5n; \\ \\ \left \{ {{8A=5} \atop {7A+B=0}} \right. \ \ \left \{ {{A=\frac{5}{8}} \atop {B=-\frac{35}{8}}} \right. \\ \\ \frac{5}{8} - \frac{35}{8 \cdot (8n+7)}](https://tex.z-dn.net/?f=%5Cfrac%7B5n%7D%7B8n%2B7%7D+%3D+A+%2B+%5Cfrac%7BB%7D%7B8n%2B7%7D%3B+%5C+%5C+%5C+%5C+A+%5Ccdot+%288n%2B7%29+%2B+B%3D5n%3B+%5C%5C+%5C%5C+%5Cleft+%5C%7B+%7B%7B8A%3D5%7D+%5Catop+%7B7A%2BB%3D0%7D%7D+%5Cright.+%5C+%5C+%5Cleft+%5C%7B+%7B%7BA%3D%5Cfrac%7B5%7D%7B8%7D%7D+%5Catop+%7BB%3D-%5Cfrac%7B35%7D%7B8%7D%7D%7D+%5Cright.+%5C%5C+%5C%5C+%5Cfrac%7B5%7D%7B8%7D+-+%5Cfrac%7B35%7D%7B8+%5Ccdot+%288n%2B7%29%7D)
![(*) \ = \frac{5}{8} - \frac{35}{8 \cdot (8n+7)}+ \frac{6}{8n+7}=
\frac{5}{8} + \frac{-35+48}{8\cdot (8n+7)}=\frac{5}{8} +
\frac{13}{8\cdot (8n+7)} (*) \ = \frac{5}{8} - \frac{35}{8 \cdot (8n+7)}+ \frac{6}{8n+7}=
\frac{5}{8} + \frac{-35+48}{8\cdot (8n+7)}=\frac{5}{8} +
\frac{13}{8\cdot (8n+7)}](https://tex.z-dn.net/?f=%28%2A%29+%5C+%3D+%5Cfrac%7B5%7D%7B8%7D+-+%5Cfrac%7B35%7D%7B8+%5Ccdot+%288n%2B7%29%7D%2B+%5Cfrac%7B6%7D%7B8n%2B7%7D%3D+%0A%5Cfrac%7B5%7D%7B8%7D+%2B+%5Cfrac%7B-35%2B48%7D%7B8%5Ccdot+%288n%2B7%29%7D%3D%5Cfrac%7B5%7D%7B8%7D+%2B+%0A%5Cfrac%7B13%7D%7B8%5Ccdot+%288n%2B7%29%7D)
НОД![(13, 8n+7)=1,13 (13, 8n+7)=1,13](https://tex.z-dn.net/?f=%2813%2C+8n%2B7%29%3D1%2C13)
![8n+7=13m; \ \ \ m=8k+3, \ \boxed{n=13k+4, \ k \in Z} 8n+7=13m; \ \ \ m=8k+3, \ \boxed{n=13k+4, \ k \in Z}](https://tex.z-dn.net/?f=8n%2B7%3D13m%3B+%5C+%5C+%5C+m%3D8k%2B3%2C+%5C+%5Cboxed%7Bn%3D13k%2B4%2C++%5C+k+%5Cin+Z%7D)
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