Problem 288: An enormous factorial

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For any prime p the number N(p,q) is defined by

N(p,q) = ∑n=0 to q Tn*pn with Tn generated by the following random number generator:

S0 = 290797 Sn+1 = Sn2 mod 50515093 Tn = Sn mod p

Let Nfac(p,q) be the factorial of N(p,q). Let NF(p,q) be the number of factors p in Nfac(p,q).

You are given that NF(3,10000) mod 320=624955285.

Find NF(61,107) mod 6110