I brute forced the answer to be 13 but I would like to know

[I brute forced the answer to be 13, but I would like to know how to do it correctly]

Solution

2^9=7 mod 101

So,

(2^9)^57=7^57=11 mod 101

2^{9*57}=11 mod 101

So, x=9*57=513

b)

7^y=(2^9)^y=2 mod 101

or 2^{9y}=2 mod 101

We know

gcd(2,101)=1

So, 2^{100}=1

2^{101}=2 mod 101

So, 2^{101+100k}=2 mod 101

and,,, 2^{9y}=2 mod 101

We are looking for integers k and y so that

101+100k=9y

Note:100=1 mod 9

And 101=2 mod 9

So, k=7 gives 101+100*7=0 mod 9

So, k=7 is one possible solution

101+7*100=801=89*9 =9y

So,y=89