Solve 14x==16 (== means "is congruent") mod 22

I thought i had to find the inverse of 14 mod 22 but now I'm confused.
Thanks!

The problem is that there is no inverse of 14 mod 22. The inverse of a mod m exists if and only if gcd(a, m) = 1. Instead, what you need to do is divide the left, the right, and the modulus by the gcd (in this case, 2). What we get is:

7x == 8 (mod 11)

This is actually equivalent to the original equation (proving that you may multiply or divide the left, right, and modulus is a simple exercise in using the definition of modulo equations). Now, gcd(7, 11) = 1, so an inverse exists. However way you calculate it, it comes to 8, since:

7 * 8 = 56 == 1 (mod 11)

Thus:

8 * 7x == 8 * 8 (mod 11)
x == 9

So, x == 9 (mod 11) is the solution. In mod 22, there are two ways to get 9 mod 11, specifically 9 and 9 + 11 = 20. So, the solution is:

x == 9, 20 (mod 22)

Hope that helps!

x =( 16/14) mod 22
x = (16 +k*22)/14 ; for some k,
x = (16+5*22)/14=126/14=9
x = (16 +12*22)/14 = 20