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I am having some trouble getting the correct answer on linear congruences. Any help would be appreciated!

Consider the following:

Solve $25x = 15 $ (mod 29)

The gdc(25,29) = 1. Therefore, x has one solution. Using the extended Euclidean algorithm we know that:

$25x_0 + 29y_0 = 1 $

We get that:

$29 = 25(1) + 4$

$25 = 4(6) + 1$

Therefore:

$1 = 25 - 4(6)$

$1 = 25 - 6(29 - 25)$

$1 = 25(7) + 29(-6)$

Multiplying by 15 we get that

$15 = 25(7*15) + 29(-6*15)$

Now $x_0 = 7*15 = 105$ which is $18$ (mod 29)

I know that my process is wrong because 18 does not satisfy the linear congruence.

Bill Dubuque
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    Note $25(18)-15 = 435 = 15(29) ; \to ; 25(18) \equiv 15 \pmod{29}$. Thus, $18$ actually does satisfy the congruence equation. Why do you think it doesn't? – John Omielan Oct 17 '22 at 04:38
  • Alternative verification of $(18)$. $(25) \equiv (-4)$ and $(15) \equiv (44).$ Then $(-4) \times (-11) = (44)$ and $(18) \equiv (-11).$ – user2661923 Oct 17 '22 at 06:06
  • Easier: $\bmod 29!:\ x\equiv \dfrac{15}{25}\equiv \dfrac{44}{-4}\equiv -11\equiv 18,$ by twiddling as explained in the linked dupe. – Bill Dubuque Oct 17 '22 at 11:42

2 Answers2

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Since $\gcd (5,29)=1=\gcd(6,29),$ we have, modulo $29,$ that $25x\equiv 15 \iff$ $ (5)(5x)\equiv (5)(3)\iff$ $ 5x\equiv 3 \iff$ $(6)(5x)\equiv (6)(3)\iff$ $29x+x\equiv 18\iff$ $x\equiv 18.$ Why did I multiply by $6$? Because when you multiply the "$5$" of the "$5x$" by $6$, you get a number that's closer to a multiple of $29$ than $5$ is. For example, modulo $29$, we have $8y\equiv 7\iff$ $32y\equiv 28 \iff$ $3y\equiv -1\iff$ $30y\equiv -10\iff$ $y\equiv -10\equiv 19.$

DanielWainfleet
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Solving a modular form equations means that you need to find the inverse of the coefficient. In particular here $$25(7)=29(6)+1$$ $$\Rightarrow 25(7)\equiv 1\quad mod(29)$$ which makes $7$ the multiplicative inverse of $25\quad mod(29)$

and given that $7(15)=29(3)+18$ we get $$25x\equiv 15\quad mod(29)$$ $$\Rightarrow 7(25x)\equiv =7(15)\quad mod(29)$$ $$\Rightarrow x\equiv 18\quad mod(29)$$

to check $$25(18)=15(29)+15$$ $$\Rightarrow 25(18)\equiv 15\quad mod(29)$$

Sam
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