No. of solutions of two equations?

Question

What is the number of distinct primes $p$ such that $$\binom{\frac{p+1}2}2\ =\ 5\cdot r\cdot q$$ where $5<r<q<p$ are primes. (See the answer given by avz2611 in the following). Similarly, if $$\binom{\frac{p+1}2}2\ =2\cdot 3\cdot\ 5\cdot r\cdot q$$ what's the number of distinct primes $p$?

Answer 1

$(p+1)(p-1)=40.r.q$ now (p+1) or (p-1) one of them have to be a multiple of 3 so you should have a multiple of 3 on the right hand side but 40 is not divisible by 3 and as both $r$ & $q$ are greater 3 and are primes even they are not divisible by 3 so no solutions possible