The linear congruence ax ≡ b(mod n) has a solution if and only if,

Consider ax ≡ b(mod n)

It is equivalent to ax - ny = b

now if d = gcd(a, n)

then there exists r, s ∈ Z such that a = dr and n = ds.

Suppose that the solution of ax ≡ b(mod n)  exists.

⇒ The solution of ax - ny = b exists

⇒ ax₀ - ny₀ = b for some x₀ , y₀ ∈ Z 

⇒ b = ax0 - ny0 

⇒ b = drx0 - dsy0 

⇒ b = d(rx0 - sy₀) where (rx0 - sy₀) is an integer.

⇒ d | b

∴ d | b where, d = (a, n)

i,e.,If d | b then the linear congruence  ax ≡ b(mod n) has d mutually incongruent solution modulo n.

Hence, the correct answer is option  3).