Fastest way to solve linear congruent equation

Question

Whats the fastest way of solving $85x=12\pmod{19}$. I can solve it but I want a quick way. I can use facts like $0=\pm19\pmod{19}$ but I am not that fast using that method.

Answer 1

$85x=12 \mod 19$ is the same as $9x=12 \mod 19$. Multiply by 2 to get $18x=24 \mod 19$ or $x=-24=14 \mod 19$

Answer 2

Hint $\displaystyle\rm\ mod\ 19\!:\ x\equiv \frac{12}{85}\equiv\frac{12}{17\cdot 5}\equiv\frac{12}{-2\cdot 5}\equiv\frac{-6}5\equiv \frac{-25}{5}\equiv \frac{-5}1$

Or: $\displaystyle\rm\ \ \,mod\ 19\!:\ x\equiv \frac{12}{85}\equiv\frac{12}{9}\equiv\frac{24}{18}\equiv\frac{5}{-1}\ \ $ (this is Gauss' algorithm)

Answer 3

Note that $85=9\ (\rm{mod}\ 19)$. So you're solving $9x=12\ (\rm{mod}\ 19)$. The fastest way would be to know an inverse of $9$ modulo $19$ (for example $17$) and multiply through by this inverse to get $x=12\cdot 17=14\ (\rm{mod}\ 19)$. Alternatively, expand the definition of a congruence to get $9x=12+19n,\ n\in\Bbb{Z}$, rewrite that as $9x-19n=12$, and solve it as you would solve a normal linear Diophantine equation.

Answer 4

First 'reduce' $85x=12\pmod{19}$ into a 'standard form':

$\tag 1 85x=12\pmod{19} \text{ iff } 9x=12\pmod{19} \text{ iff } 3x=4\pmod{19}$

The smallest positive integer that solves $3x=4\pmod{19}$ is found by taking these two steps:

• $\quad \text{Let } k \ge 1 \text{ be the smallest integer such that } 3 \text{ divides } 4 + 19k$

• $\quad \displaystyle \text{Set } x = \frac{4 + 19k}{3}$

Does $3$ divide $23$? No.
Does $3$ divide $42$? Yes and $\frac{42}{3} = 14$.

$\tag {ANS} x \equiv 14 \pmod{19}$