A friend thinks if he performs the following steps, he will develop a new distribution called the World's Greatest distribution. Suppose $X_1,X_2, \ldots ,X_n$ are iid $N(\mu, \sigma^2)$. He takes each one of these random variables, subtracts $\mu$ from each of them, and divides each of them by $\sigma^2$. He then squares each one of these new random variables, takes their sum and makes a new random variable he calls $D$. Then he adds $4$ $\mathrm{Gamma}(3,2)$ random variables, $4$ $\mathrm{Weibull}(1,2)$ random variables, and $9$ $\mathrm{Exponential}(2)$ random variables to $D$ where all of these random variables including $D$ are mutually independent. He calls this new random variable $U$. Now he divides $U$ by a constant $C$ and then divides that result by a chi-squared random variable with $100$ degrees of freedom divided by $100$. This chi-squared random variable is independent of $U$. He calls this new random variable $M$. After doing all of this work, he then takes the reciprocal of $M$ and calls this final random variable $B$. Your job is to check his work and see if this resulting random variable is something you know already. You find out that it is. Thus, you go and tell him the news. What distribution is it? (Note: You will need to determine the constant $C$ that divides $U$.)
For this I see that I have $N$ normal distributions and multiply by the moment generating functions of the $4$ gammas, $4$ Weibulls and $9$ exponentials which means I only add up the exponents because mgf is of the same form. Thus I got $\bigl(\frac{1}{1-2t})^{\frac{n}{2}+25}$. Getting a little lost after that.