I was reading this paper by Erdos in which he derives that there exists infinitely many $n$ such that for any primes $p,q $ where each prime is greater than $2$
$$\text{gcd}\left(\binom{2n}{n},pq\right)=1$$
This is the page containing theorem 2 in that paper.
He says that from $(1)$
$$A(p;x)\leq A(p;(t+1)p^{r})\leq (t+1)\left(\frac{p+1}{2}\right)^{r}$$
Fact $(1)$ is,
$\textbf{Question}$:- I understood the part where $A(p;x)\leq A(p;(t+1)p^{r})$ but why is $$A(p;(t+1)p^{r})\leq (t+1)\left(\frac{p+1}{2}\right)^{r}$$ ??