The number of positive integers which divide (1890)*(130)*(170) and are not divisible by 45 is ____________.

Let N = (1890)*(130)*(170)

N = $$\left(2\times3^3\times\ 5\times\ 7\right)\times\left(2\times5\times13\right)\times\ \left(2\times5\times17\right)$$

N = $$2^3\times3^3\times5^3\times7\times13\times17$$

We did the prime factorisation of N.

The number of factors of a number N with prime factorisation  N = $$p^a\times q^b\times\ r^c$$

Number of factors = $$(a+1)(b+1)(c+1)$$

Therefore, the number of total factors of N = $$2^3\times3^3\times5^3\times7\times13\times17$$

Number of factors = (3+1)(3+1)(3+1)(1+1)(1+1)(1+1) = 512

We will calculate the factors of N that are divisible by 45

N = $$3^2\times5\ \left(2^3\times3\times5^2\times7\times13\times17\right)$$

Factors of N that are divisible by 45 = (3+1)*(1+1)*(2+1)*(1+1)*(1+1)*(1+1) = 192

We need to find the number of positive integers which divide (1890)*(130)*(170) and are not divisible by 45 is

Factors of N - Factors of N divisible by 45 = 512-192 = 320