Some examples of consecutive numbers where first is a multiple of 2, the second is a multiple of 3 and the third is a multiple of 4:
14, 15, 16
26, 27, 28
38, 39, 40
Here is my method:
Find the L.C.M. of the three (or any other number) consecutive numbers whose multiples we are looking for. In the above scenario, the L.C.M. of 2,3,4 will be 12.
Now, if we add L.C.M. of these three numbers to any of the numbers, the result must be a multiple of that number and hence we can add the L.C.M. to the three numbers to get three consecutive numbers that are multiples of 2,3 and 4 respectively.
Using the same logic,
For four consecutive numbers which are multiples of 2,3,4,5:
62, 63, 64, 65 (as L.C.M. is 60)
For five consecutive numbers which are multiples of 2,3,4,5,6:
62, 63, 64, 65, 66 (as L.C.M. is 60)
And for the final question, the L.C.M of 1,2,...,10 will be 2520. Hence the numbers will be:
2521, 2522, 2523, 2524, 2525, 2526, 2527, 2528, 2529, 2530