Idea : work with powers of primes

Rank(A) mod 7 = 3 = rank(A)

Rankq(A) = greatest i such that q does not divide si

If all the primes appearing in the Smith form are known

Then compute the ranks modulo their powers

Rank(A) mod 3 = 1

Rank(A) mod 9 = 3

Rank(A) mod 2 = 2

Rank(A) mod 4 = 3