To find sets of consecutive numbers which would also be multiples of 2,3,4, I listed all the numbers:
2,3,4 - Yes
4,6,8 - No
8,9,12 - No
10,12,16 - No
12,15,16 – No
14,18,20 – No
14,15,16 - Yes
I noticed that the difference between all the numbers was 12 – 14-2, 15-3, 16-4.
We know that if you add a multiple of a number to the number the answer will always be a multiple for eg: if you add 6 which is a multiple of 3 to 3 it will give us 9 which is also a multiple of 3 or for 5 – 5+25=30, 30 is a multiple of 5.
So, if you add a common multiple of numbers the result should be a multiple of those numbers, like here I found 12. I checked if 12 is a multiple of all the numbers:
2x6=12
3x4=12
4x3=12
So, I added 12 again to 14,15,16 to check if I will get consecutive numbers – 26,27,28. Here also, 26 is a multiple of 2, 27 is multiple of 3 and 28 is a multiple of 4.
So, if you add the same number to a set of consecutive numbers, you will always get a new set of consecutive numbers.
12 works because it can be divided by 2, 3 and 4.
It is also the least common multiple (LCM) – 2x3x2=12
I tried this again for 3,4,5. The LCM for 3,4,5 will be 3x2x2x5=60 and after adding I found these sets:
3,4,5
63,64,65
123,124,125
I used this to find the least common multiple between different sets to find out consecutive numbers which are also multiples.
For finding sets of ten consecutive numbers which are multiples of 1-10:
The LCM of 1,2,3,4,5,6,7,8,9,10 is
2x3x2x7x2x3x5 = 2520
So, the sets will be:
1,2,3,4,5,6,7,8,9,10
2521,2522,2523,2524,2525,2526,2527,2528,2529,2530
5041,5042,5043,5044,5045,5046,5047,5048,5049,5050
7561,7562,7563,7564,7565,7566,7567,7568,7569,7570