For example. given $15x\equiv 10\ (mod \ 16)$, what method would someone use to reduce this to $x \equiv 6\ (mod \ 16)$?
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4Take the negative of both sides. – David Apr 05 '18 at 07:22
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$15 = 3\cdot 5$. So find the multiplicative inverse of $3$ which is $11$ and of $5$ which is $13$. Then multiply $10 \cdot 11 = 110$. Modulo $16$ that's $110 - 6 \cdot 16 = 14$. Then $14 \cdot 13 = 140 + 42$. Modulo $16$ that's $12 + 10 = 22 = 6 \pmod{16}$.
But that takes a while.
So notice that $15 = 16 - 1 = -1 \pmod {16}$ and that $-10 = 16 - 6$. So you multiply both sides by $-1$.
Daniel Donnelly
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I am kind of confused how taking the negative leads anywhere. You now have $-15x \equiv -10(mod 16)$ but I don't see how that changes anything exactly – user3491700 Apr 05 '18 at 09:41
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@user3491700 $-y = 16 - y \pmod {16}$. Use that to transform the 15 and the 10. – Daniel Donnelly Apr 14 '18 at 02:12