Question

Solve the congruence 5x≡1(mod12)

Hint: 0≤x≤11

Answer 1

$ax ≡ b(mod \space m)$

Step 1

GCD of a and m

GCD of 5 & 12 

since 5 is a prime , so the GCD is 1

step 2

1/1=1

So, solutions exists

step 3

$5x ≡1+12n(mod \space 12)$

where n is a natural number

finding n such that (1+12n) is divided by 5

if n= 1

$5x ≡1+12(mod \space 12)\\ 5x ≡13(mod \space 12)$

for n=2

$5x ≡1+12*2(mod \space 12)\\ 5x ≡25(mod \space 12)\\ x ≡5(mod \space 12)$

So, the solution of n between 0 and 11 will be 5. In general form it will be x = 5+12m