Multiplicative modulo inverse

#1

why 1 is considered as gcd of (a,b).
and then it is considered as the essential condition for exisatace of multiplicative modulo inverse

#2

Some Game Theory has proved the fact that:
The multiplicative Modular inverse of “a modulo b” exists if and only if a and b are relatively prime (i.e., if gcd(a, b) = 1).

if u urself cite some example:
a = 3 b = 9
a%b == 0
u can never find the inverse modulo
because there will be no ‘m’ for which ( a * m ) % b == 1

& a = 4 and b = 7
a% b = = 1
so here we can find a modulo inverse,

u could try out with other example , where gcd of 2 no if any thing other than1 , you wont be able to find mutiplicative modulo