There are various definitions of the parameters of a gamma distribution, but there are always two parameters, and since you know (given the answer) both the mean and the variance, you have two equations, and these allow you to identify the parameters and thus the distribution.

For instance, if you define a "Gamma distribution with scale parameter and index " to be the distribution with density on , then you can compute the mean (it is ) and the variance (it is ), so that you must have and .

Solving this, you get and . This answers your question. By the way, the "index" you were given happens to be the "index" of my definition. This could have allowed to find from the only knowledge of the mean.