Cambria was building various cuboids from \(1\times 1\times1\) cubes. She initially built one cuboid, then increased its length and width by \(1\) and reduced its height by \(2\). She then understood that she needs the same number of \(1\times 1\times 1\) cubes to build both the original and new cuboids. Prove that the number of cubes used for each of the cuboids is divisible by \(3\).