Is there a direct way to determine the number of positive factors that a given number like 1800 has?
The easiest way is through easy websites such as this:
Alternatively, you can use prime factorization. In this case,
1800 = 2*2*2*3*3*5*5. Thus any divisor contains at most three
factors of 2;
that means there are four possibilities: 0, 1, 2, or 3 factors
of 2. Similarly there are three possibilities for the number of
factors of 3 in a divisor, and the same for factors of 5.
Multiplying these together, we find there are 4*3*3 = 36 positive
integer divisors of 1800.
In general, take the exponents in the prime factorization, add
one to each of them, and multiply the resulting numbers