Suppose the required number is N, and that it has prime decomposition:

N=p_1^(n_1) p_2^(n_2), ... , p_m^(n_m).

where n_1, n_2 ..are all >=1. Then the number of divisors this number has (including 1 and N) is:

d=(n_1+1)(n_2+1)...(n_m+1).

Now if d=75, as 75=3.5^2, any factorisation of d has at most 3 factors (they

are 3, 5, 5). Also we already know that two of the p_i 's are 3 and 5, the

smallest N with the required properties has 2 as a factor with a multiplicity

as large as can be made consistent with the conditions on N, so:

Making the correction Plato pointed out:

N=2^4 3^4 5^2 = 32400.

RonL