This one ..well I hope it's ok ..bit rusty on the mods.

To find the reciprocal of 3 (mod 22) we must first ensure it exists by finding if they have 1 as their greatest commonfactor(thanks HallofIvy) remembered to put this

gcd(22,3): 22 = 7 ⋅ 3 + 1 1 = 22 ⋅ 1 – 3 ⋅ 7

-7 ⋅ 3 ≡ 1 (mod 22)

3x ≡ 5 (mod 22)

-7 ⋅ 3x ≡ -7 ⋅ 5 (mod 22)[multiply both sides by -7]

1x ≡ -35 (mod 22) [sub in 7 ⋅ 3 ≡ 1 (mod 22)]

x ≡ -13 (mod 22) [add 22 to -35]

x ≡ 9 (mod 22)[add 22 to -13]

Which I’m pretty sure works because 5 mod 22 ≡ {...,-39,-17,5,27,49,71, ...}

So you just find one that 3 goes into (27) to get 3x9 ≡ 5 mod 22, so x ≡ 9 (mod 22)

Is this ok??