The greatest common divisor  31C3,31C5…,31C29 is

We have, for 3≤r≤29r!(31−r)! 31Cr=31!As the prime 31 divides R.H.S. and 31 does not divide r! and(31−r)! for β≤r≤29, we get 31∣ 31CrAlso, Since  31C29=(31)(3)(5),31C3=(31)(29)(5) and 31C5=(31)(29)(7)No prime other than  31  can divide all the numbers.Thus greatest common divisors of the given numbers is  31 .