Hamiltonian Cycles on a Torus

Knuth教授のThe Art of Computer Programming volume 2, fascicle 4 の問題に関して悩んでいます。
The problem requires a proof that the number of Hamiltonian cycles on an m x n torus is:
d-1
SIGMA(d CHOOSE k) where the gcd*1^t. Multiplying both sides by beta^t yields beta^(t+d)=1, so t+d = 0 mod n.
The next step shows that t+k is relatively prime to c if and only if (d-k)m/d is relatively prime to kn/d.
1. I am not sure how this is derived.
2. The proof concludes here, and I am not sure where the number of Hamiltonian cycles as the sum of (d choose k) is inferred.
3. Note that the torus is a torus in "Cm x Cn", where Cm is the cyclic group of m elements (0, 1, . . . , m-1).

*1:d-k)m,kn)=d k=1 Here d is the gcd of m and n. Note that there are two "generating" arcs here, alpha and beta. Alpha takes (x,y) to ((x+1) mod m, y), and beta takes (x,y) to (x, (y+1)mod n). There are mn vertices, 0<=x