Is there a way to figure out how many divisors are in the numbers 1 million without writing them out?

I am very confused, does anyone know how to figure out the answer to this problem? I know that there must be a way to solve using a formula or something!Is there a way to figure out how many divisors are in the numbers 1 million without writing them out?
Yes, there is a way to solve this problem using the prime factorization of 1 million. You should use the following function which is usually denoted by the Greek letter tau:





tau(n) = the number of divisors of n (including 1 and n).





So, for instance, if p is a prime, then it's only divisors are 1 and p. So, tau(p) = 2.





Here's how it is useful: if m and n have no common factor other than 1, then tau(m*n) = tau(m) * tau(n).





(The reason why is that if d is a divisor of m*n, then there are factors a and b of d so that d = a*b, a divides m, and b divides n. Since m and n have no common factor, the factors a and b are unique. Conversely, if a divides m and b divides n, then a*b divides n.)





So, for instance, since 24 = 3*8 and 3 and 8 have no common factor other than 1, tau(24) = tau(3) * tau(8).





The last fact you need is that tau(p^k) = k+1 if p is a prime.


(The only divisors of p^k are 1, p, p^2, ..., p^k.)





So, for instance, tau(24) = tau(3) * tau(8) = tau(3) * tau(2^3) = 2*4 = 8.





Here's one last example: tau(1620) = tau(2^2 * 3^4 * 5) = tau(2^2) * tau(3^4) * tau(5) = 3*5*2 = 30.





The limitation of this method is that you need to figure out the prime factorization of the number first. For the number 1 million this is not so bad; just write out some smaller powers of ten and you'll see what to do. In general, however, it is hard to factor a number into primes.