Problem #PRU-66397

Problem

Cambria was building various cuboids from $1\times 1\times 1$ 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 $3$.