The number of positive integers that divide \((1890) \cdot (130) \cdot (170)\) and are not divisible by \(45\) is

There are $320$ eligible divisors. The product is $2^3\cdot3^3\cdot5^3\cdot7\cdot13\cdot17$, giving $4^3\cdot2^3=512$ divisors in all. A multiple of $45$ has two choices for the exponent of $3$, three for the exponent of $5$, and $4\cdot2^3$ choices for the remaining exponents, so subtract $192$ from $512$. Reporting $192$ counts the excluded divisors.