Let $X, Y,$ and $Z$ be the sets of positive divisors of $10^{60}, 20^{50}$ and $30^{40}$ respectively. Find $n (X\cup Y\cup Z) $.

Question

Let $X, Y,$ and $Z$ be the sets of positive divisors of $10^{60}, 20^{50}$ and $30^{40}$ respectively. Find $n (X\cup Y\cup Z) $.

I've trying to solve this question since long time but I am unable to do so. I have tried to use Venn diagrams but such approaches did not help me. I am not good at combinatorics so therefore I am seeking help? Would someone please help me to solve this question?

Thanks for help!

Answer 1

Note that $10^{60} = 2^{60} \times 5^{60}$ , $20^{50} = 2^{100} \times 5^{50}$ and $30^{40} = 2^{40} \times 3^{40}\times 5^{40}$.

Divisors of the form $ 2^{i} \times 5^{j} $ are shown in the diagram below.

And the number of factors is indicated in each rectangle.

The other factors have the form $ 2^{i} \times 3^{j} \times 5^{k} $ where $i=0,1,\cdots,40$ ,$j=1,2\cdots,40$ and $k=0,1,\cdots,40$.

Now we just need to do the arithmetic $610+3111+2040+ 41 \times 40 \times41=?$.