Explanation:

3⁹⁰ + 5⁹⁰ can be written as (3²)⁴⁵ + (5²)⁴⁵ = (9)⁴⁵ + (25)⁴⁵
Any number of the form aⁿ + bⁿ is a multiple of (a + b) whenever n is odd.
So (9)⁴⁵ + (25)⁴⁵ is a multiple of 9 + 25 = 34
So, the remainder when we divide (3²)⁴⁵ + (5²)⁴⁵ by 34 is equal to 0.