Problem 610

We start by finding the prime factors of 1184: $1184=2 \cdot 592=2^{2} \cdot 296=2^{3} \cdot 148=2^{4} \cdot 74=2^{5} \cdot 37$. The positive divisors of 1184 cannot contain prime factors other than 2 and 37, and cannot contain more than 5 factors of 2 or 1 factor of 37. Thus, the positive divisors are $1,2,4,8,16,32,37,74,148,296,592,1184$. The sum, $S$, of these divisors is $S = 1+2+4+8+16+32+37+74+148+296+592+1184 = (1+2+4+8+16+32)+37 \cdot(1+2+4+8+16+32) = (1+2+4+8+16+32) \cdot(1+37) = 63 \cdot 38 = 2394$.