Explanation:

Any factor of 2¹³ * 3⁶ * 5⁹. will be of the form 2^p * 3^q * 5^r.
When we are trying to find the number of factors without any constraints, we see that.
p can take values 0, 1, 2, 3......13 – 14 values.
q can take values 0, 1, 2......6 – 7 values.
r can take values 0, 1, 2, 3......9 – 10 values.
So, the total number of factors will be 14 * 7 * 10.
This is just a rehash of our formula (a + 1) (b + 1) (c + 1).
In this scenario we are looking for factors of 213 * 36 * 59 that are multiples of 120.
These will also have to be of the form 2p * 3q * 5r. But as 120 is 23 * 3 * 5, the set of values p, q, r can take are limited.
p can take values 3, .......13 – 11 values
q can take values 1, 2, .......6 – 6 values
r can take values 1, 2, 3, .......9 – 9 values
So, the total number of factors that are multiples of 120 will be 11 * 6 * 9 = 594.