Our solution works because the numbers in the adjacent cells to the multiples number have a product that equals the number in the middle. In addition, the Sudoku has no number that is the same in any row, column or 3x3 cell. Therefore, we have completed the Sudoku.
To start with, we listed all of the factors that the numbers in between two cells, provided they were both between 1 and 9 inclusive. We started with numbers with only one pair of factors, like 20, 21 and 54. This was made easier when we realised that some numbers that had two different factor possibilities could only have one solution because of all of the numbers around them that we had already filled in; some wouldn't work with the rest of the puzzle's rules. The cell with 8 in the middle could be 2 and 4 or 1 and 8. However, 2 was already in the same line.
Once we had all of the numbers from the multiples, we could work it out like a normal Sudoku. The numbers from the multiples made it quite easy to complete when we had started using it like a real Sudoku.