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This seems more like a homework question than something about Sage. Sage won't be able to give you a formula for the number of zero divisors of Z mod (pqr) (although one exists); you need some more theoretical knowledge for that. However, here is a naive function which counts the number of zero divisors of Z mod n:

def number_of_zero_divisors(n):
    zero_divisors = 0
    for i in range(1, n):
        for j in range(1, n):
            if (i*j) % n == 0:
                zero_divisors += 1
                break
    return zero_divisors

Here is some sample output:

sage: number_of_zero_divisors(2*3*5)
21
sage: number_of_zero_divisors(5*7*11)
144