Product 1

$P=\prod_{r=4}^{50}[\frac{r-3}{r+1}]=\red{\prod_{r=4}^{50}[r-3]}\times\blue{\prod_{r=4}^{50}[\frac{1}{r+1}]}$ Substitute $x=r-3$ in the $\blue{first\space product}$ and $y=r+1$ in the $\red{second\space product}$ $\Rightarrow P=\prod_{x=1}^{47}[x]\times\prod_{y=5}^{51}[\frac{1}{y}]$ $=\prod_{x=5}^{47}[x]\times\prod_{y=5}^{47}[\frac{1}{y}]\times1\times2\times3\times4\times\frac{1}{51}\times\frac{1}{50}\times\frac{1}{49}\times\frac{1}{48}$ $=2\times3\times4\times\frac{1}{51}\times\frac{1}{50}\times\frac{1}{49}\times\frac{1}{48}=\frac{1}{249900}$ As $1\in\mathbb{Z}^+$ , $249900\in\mathbb{Z}^+$ and $\gcd(1,249900)=1$ , therefore the answer is $=249900+1=249901$