Fermat's little theorem .

Fermat's little theorem tells us that if $\gcd(a, p)=1$ , then $a^{p-1}\equiv 1\pmod p$ [the $p$ here is a prime number].

So, $1^{96}+2^{96}+3^{96}+\cdots +96^{96}\equiv 1+1+1+\cdots 1\equiv 96 \pmod {97}$ .

And our answer is $\boxed{96}$ .

Python code:

s=0

for i in range(96):

a=i+1

for j in range(96):

    a=a%97

    if j==93:

        a=a

    else:

        a=a*(i+1)

s=s+a

print s%97

By Fermat's little theorem, we know that $a^{96}$ mod $97$ ≡ 1. Therefore $1^{96} + 2^{96} + 3^{96} + ... + 96^{96}$ mod $97$ ≡ 96.

Java Code;

public class Brilliant{

public int SampleMethod(){

int sum= 0;

int u = 0;

for(int x = 1 ; x<=96;x++){

sum += u;

for(int f= 1; f<=96;f++){

u = x*x;

}

}

return sum% 97;

}

}

Sorry, I dont know how to create this code block.