If \(n = 2^{10} \cdot 3^{14} \cdot 5^{8}\), how many of the natural-number factors of \(n\) are multiples of 150?
I got 702, but that was wrong...
I'm not the best at this, but this is my attempt:
150 = 2 x 3 x 5 x 5 -- so every multiply must have at least 1 two, 1 three, and 2 fives.
So, from the 210 (which consists of 10 twos),
you can select either 1, 2, 3, 4, 5, 6, 7, 8, 9, or 10 twos -- 10 different selections.
From 314 (which consists of 14 threes),
you can select either 1, 2, 3, ... 14 threes -- 14 different selections.
From 58, you must select at least 2 fives, but it could also be 3, 4, 5, 6, 7, or 8 fives
-- 7 different selections.
Multiplying 10 x 14 x 7, I get 980 different ways.