Can someone help me with $109x\equiv 24\pmod{26}$? I don't really know what to do. Thanks for the help.
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Look up the questions on $ax\equiv b\bmod m$ here at this site! See for example here. What have you tried ? – Dietrich Burde Feb 27 '20 at 20:35
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i tried bashing it by adding 26 to 24 to get a multiple of 109. – Dummy Dumb Feb 27 '20 at 20:37
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$109\equiv 5\pmod{26}.$ $24\equiv 50\pmod{26}.$ – Thomas Andrews Feb 27 '20 at 20:38
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Welcome to Mathematics Stack Exchange. Modulo $26$, you may add or subtract multiples of $26$ – J. W. Tanner Feb 27 '20 at 20:39
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$109x = 24 + 26y$.. $(426+5)x = 24+26y$.. $5x = 24+26(y-4)$..$5x=4+45+26(y-4)$..$5(x-4)=4+26(y-4)$.. $5(x-4)=30+26(y-5)$..$x-4=6+\frac{26(y-5)}5$. That's an integer but $5$(prime) doesn't divide $26$ so $5|y-5$ (BTW we don't give a fig about what $y$ is)..$x-4=6+26\frac{y-5}5$.. $x =10 +26\frac{y-5}5$..$x\equiv 10\pmod 16$..Verify: $10910\equiv 1090\equiv24+4126\equiv 24\pmod{26}$. – fleablood Feb 27 '20 at 22:12
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"i tried bashing it by adding 26 to 24 to get a multiple of 109" Good Idea. But notice that $109 > 26$ so try bashing that down to $109=104 + 5$ so $109x\equiv 5x \equiv 24\equiv 24+26\equiv 50\pmod {26}$. Now DON'T assume you can do division, but note that if $x \equiv 10\pmod{26}$ then $510\equiv 50 \equiv 24\pmod 26$. Or $10910\equiv1090=41*26 + 24\equiv 24\pmod{26}$. – fleablood Feb 27 '20 at 22:16