solve the following linear congruence 14x 35 mod 49Solutiong

solve the following linear congruence

14x 35 mod 49

gcd (14, 49) = 7

since 7 is a divisor of 35, so there will be a solution of this equation.

14x = 35 mod 49

is equivalent to

14x = 35 + 49q

where x and q are integers.

2x = 5 + 7q

2x = 5 mod 7

number congruent to (1 mod 7) is = 8, 15, 22, etc and -6, -13, -20 etc.

Now we see that 2 is a factor of 8, so we multiply both side of congruence to (8/2 = 4)

2*(4) = 8 = 1 mod 7

so we have

8x = 20 mod 7

8x = 6 mod 7

so solutions will be

x = 6, 13, 20, 27, 34, 41, 48