If k = abc + 1, then prove that (k,a) = (k,b) = (k,c) = 1?

I have gotten this far, can anybody help me with the last few steps:

Suppose d divides k and d divides a, then k = du and a = dv

So du = k = abc + 1 = dvbc + 1

Therefore du = dvbc + 1......how do I show that d divides 1?
Update: The parentheses imply greatest common divisor
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