Question

If a and b are positive integers with no common factor

Answer 1

a and b are integer but It may be negative

Answer 2

If x,y∉Z and x+y∈Z, then ⌊x⌋+⌊y⌋=x+y−1. Therefore,

⌊jab⌋+⌊(b−j)ab⌋=a−1(1)

If (a,b)=1, then for 0<j<b, we have jab∉Z.

Summing (1), we get

∑j=1b−1⌊jab⌋+⌊(b−j)ab⌋=(a−1)(b−1)(2)

(2) counts each term in the sum we want twice, so we get

∑j=1b−1⌊jab⌋=(a−1)(b−1)2(3)

Hope this helps ☺️☺️☺️