prove that (2×4×6×8×....2016)-(1×3×5×...2015)is divisible by 2017

If a number X is divisible by Y, then the remainder of X after division by Y is zero. Calculating the remainder after division can be greatly simplified, you often don't need to actually divide X by Y to calculate the remainder.

The remainder after division of X by Y is denoted as X Mod Y. It's clear that subtracting any multiple of Y from X won't change the remainder:

X Mod Y = (X - k Y) Mod Y

It follows from this that multiplying a number of terms and then taking Mod Y yields the same result as reducing each term Mod Y and then multiplying these reduced terms and then taking Mod Y again at the end.

Let's consider rewriting:

(2×4×6×8×....2016) Mod 2017

We are then allowed to subtract 2017 from each factor in the product, this yields:

(-2015)x(-2013)x...x(-1) Mod 2017 =

(1×3×5×...2015) Mod 2017

because you have 1008 minus signs that cancel out.

Then this is the same as taking Mod 2017 of the last term, so the two terms subtracted from each other yields zero.