Find all solutions to the congruence 6x≡9(bmod45)

Question

Find all solutions to the congruence  6x≡9(bmod45)

unessodopunsep · Accepted Answer

Step 1
To Determine :-
All solutions to the congruence :

6 x \equiv 9 (b mod 45)

Step 2
All solutions to the congruence :

6 x \equiv 9 (b mod 45)

Here , in this question ,
The general linear congruence is

a x \equiv b (b mod m)

On comparing we get,
a=6, b=9, m=45

∴ d = (a, m) = (6, 9) = 3

.
Now, d(=3) divides b (=45).
Thus , the solution of given congruence exists and is unique.
Now,

6 x \equiv 9 (b mod 45)

2 x \equiv 3 (b mod 15)

Then,
2x-3=15k
2x=15k+3

\Leftrightarrow x = \frac{15 k + 3}{2}

Substitute, k = 1 , 3 , 5, we get

x = \frac{15 \times 1 + 3}{2} = \frac{15 + 3}{2}

\Rightarrow x = \frac{18}{2} = 9

x = \frac{15 \times 3 + 3}{2} = \frac{45 + 3}{2}

\Rightarrow x = \frac{48}{2} = 24

x = \frac{15 \times 5 + 3}{2} = \frac{75 + 3}{2}

\Rightarrow x = \frac{78}{2} = 39

Hence , All The solutions of the congruence :
x=9,24,39

Find all solutions to the congruence 6x\equiv 9(\bmod 45)