Question 1210186
<br>
Note that 6^6, 10^10, and 15^15 are all factors of 30^30.  It is therefore only necessary to find the number of positive integers that are factors of 30^30.<br>
The standard method for determining the number of factors of a given integer N is
(1) find the prime factorization of N;
(2) add 1 to each exponent in the prime factorization; and
(3) multiply the numbers from step (2)<br>
The prime factorization of 30 is 2*3*5, so the prime factorization of 30^30 is (2^30)(3^30)(5^30).<br>
The number of factors of 30^30 is (30+1)(30+1)(30+1) = 31^3 = 29791<br>
ANSWER: 29791<br>