If a and b are both divisible by d, then a and b are *multiples* of d. Then there are two integers, k and L, such that dk = a and dL = b.

Step 1: Here, they are just substituting dk for a and dL for b.

> ax + by = (dk)x + (dL)y

Step 2: Looking at the right hand side of the equation above, factor out the d.

> dkx + dLy = d(kx + Ly)

> So ax + by = d(kx + Ly)

Step 3: Since multiplying two integers together gives you an integer, kx and Ly must both be integers. And since adding two integers together gives you an integer, we also know that kx + Ly is an integer as well. This means that d(kx + Ly) is an integer that is obviously divisible by d.

Here, they say this by introducing a new integer called k’ such that k’ = kx + Ly.

> d(kx + Ly) = dk’

> So ax + by = dk’ which means that ax + by is also divisible by d.