For the first section, I wrote out the numbers '2,3 and 4'-the obvious solution. I then searched for three more sets: '14,15,16';'26,27,28' & '38,39,40'. Once I found them, I noticed how each time: the tens digit of the first number increased by 1 every time, and also how the units digit of each first number was the same as the units digit of the last number in the previous set. I went on to figure out that if you write them out in columns:
2 3 4
14 15 16
26 27 28
38 39 40
50 51 52
62 63 64
how each number is 12 more than the above. Therefore, as 2,3 and 4 are all factors of 12, this is the number the columns increase by to make various sets of consecutive numbers.
For the numbers 3,4 and 5, I did the column strategy again, and figured out how the same thing happens, but with the number 60, as that is their Lowest Common Multiple (LCM).
3 4 5
63 64 65
123 124 125
183 184 185
For 4,5 and 6, again I used the columns, and found out that they also increase by 60, and are therefore similar to 3,4 and 5.
4 5 6
64 65 66
124 125 126
184 185 186
Then, to work out 2,3,4 and 5, I combined my results; and figured out that as 2 divides into 4, their LCM must again be 60. To work out these sets, you simply write out '2','3','4' and '5', then add sixty to each number.
Finally, to work out 2,3,4,5 and 6, it is once again 60! This is another combination of the first and third sets, but because two divides into 4 or 6, and 3 divides into six, it once again becomes and LCM of 60. Therefore, you use a similar strategy to the previous set, and write out '2,3,4,5 and 6', then add 60 to each one!
So, in conclusion, this puzzle was all about Lowest Common Multiples. This strategy could possibly be employed on different consecutive numbers, too.