I understand this question has been asked a couple of times, but from what I am seeing, the parameters shown are different so I am not sure if I am on the right track.
My question goes like this: Given X has a gamma distribution with parameters $\alpha$ and $\theta$, then Y = cX has a gamma distribution with parameters $\alpha$ and c$\theta$.
My approach was to use the cdf such that P(cX $\leq$ y),which results in F$_{x}$($\frac{y}{c}$). But after expanding it out, I do not know how to substitute into the gamma function, which requires integration.
If $X$ has Gamma-distribution with parameters $\alpha$ and $\theta$ then its PDF on support $(0,\infty)$ takes the form:$$f_X(x)=\frac1{\Gamma(\alpha)\theta^{\alpha}}x^{\alpha-1}e^{-\frac{x}{\theta}}$$and is (as usual) the derivative of the corresponding CDF.
Let $c$ be a positive constant.
Taking the derivative of $F_{cX}(x)=F_X(\frac{x}{c})$ we find by taking the derivative and applying the chain-rule:$$\frac1cf_X\left(\frac{x}c\right)=\frac1{\Gamma(\alpha)\eta^{\alpha}}x^{\alpha-1}e^{-\frac{x}{\eta}}$$on support $(0,\infty)$ where $\eta:=c\theta$.
This is evidently the PDF of Gamma-distribution with parameters $\alpha$ and $\eta=c\theta$.