Why x * i / j is not equal to x / j * i ?

Revision en2, by dreamoon_love_AA, 2019-05-13 15:50:44

Hi, all.

I found a magic thing that $$$x\ /\ i * j$$$ is not always equal to $$$x * j\ /\ i$$$! For example, when $$$x = 12, i = 9, j = 6$$$, then $$$x\ /\ i * j = 6$$$ but $$$x * j\ /\ i = 8$$$. The fact surprised me.

Now, given three positive number $$$x$$$, $$$a$$$, $$$b$$$, I wonder how may pair of $$$(i,j)$$$ satisfy $$$x\ /\ i * j = x * j\ /\ i$$$ and $$$1 \le i \le a$$$ and $$$1 \le j \le b$$$?

I hope I can solve about $$$20$$$ tests which satisfy $$$1 \le x,a,b \le 10^9$$$ in 1 second. Can you help me?

PS. This is a problem that I give in some local contest. I feel it's interesting and share it here.

PS2. I think it's a math problem instead of programing problem.

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en2 English dreamoon_love_AA 2019-05-13 15:50:44 179 (published)
en1 English dreamoon_love_AA 2019-05-13 15:44:39 508 Initial revision (saved to drafts)