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To begin with I used the simple 2, 3, 4 answer to solve the question of: Can you find three consecutive numbers where the first is a multiple of 2, the second is a multiple of 3 and the third is a multiple of 4? This allowed me to use it as a base for other workings. Next I used the last number in the sequence, 4, to increase the units by. After 3 attempts at this I came to the sequence of 14, 15, 16 which once again works for the question. As maths is made up of patterns I figured that it would be no different for this and attempted to find a pattern within the numbers. As is immediately obvious, the numbers go up in twelves: 14 – 2 = 12, 15 – 3 = 12 and 16 – 4 = 12. This might work a second time. As a matter of fact – it did – the next sequence was 26, 27, 28. However, there must be something more than the fact that it is going up in twelves.
12 is in fact the lowest common multiple of the numbers 2, 3 and 4 which brings forward the idea that the theory of: the numbers are going up in the lowest common multiple of the sequence of numbers used in the question.
In order to reinforce this theory, I used it on the second series of numbers, 3, 4, 5. The lowest common multiple of this sequence is 60 which in effect causes the next set to be 63, 64, 65. Once again, it has worked. Now for 4, 5, 6 (the lowest common multiple is 60 again) we are left with the sequence of 64, 65, 66 which has proven my theory for a set of 3 numbers.
Now to try 4.
Considering the fact that my theory worked for a sequence of three numbers, I have predicted that it will also work with four. My base for this test is 2, 3, 4, 5 which again has a lowest common multiple of 60. Consequently, if my theory is correct, the following sequence will be 62, 63, 64, 65. This does correspond correctly with the question which proves that my theory works for a four number sequence.
Now to test 5.
My baseline sequence is now 2, 3, 4, 5, 6 which has in fact got a lowest common multiple of 60 – again. As a result the second answer to this question involving five different numbers is 62, 63, 64, 65, 66.
In conclusion, my theory was right from almost the very start and it has been proven to work for not only three numbers but four numbers and five numbers. If I were to continue this test I would see if it worked for six numbers and seven numbers but it seems as though it would be unnecessary because of our former results. Overall it has been shown that for sequences of 3, 4 or 5 numbers the difference between each of the separate answers to the question is the lowest common multiple of the series of units.