Breaking a small-scale Example Diffie-Hellman?
Actually, given $p,g,g^x\bmod p,g^y\bmod p$ deciding whether a given number is actually $g^{xy}\bmod p$ is also known as the Decisional Diffie-Hellman Problem (DDH), which sits at the basis of the (formal) security for the ElGamal cryptosystem and is assumed to be hard for properly chosen values of $p,g$ and $x,y$ in general.

For your particular scenario the best solution is to "just" solve the discrete logarithm problem, ie computing $x$ or $y$ and then computing $g^{xy}\bmod p$ to see which number falls out (113).