I have found that all multiples of three can be written as a sum of three consecutive numbers. I have found a pattern, as follows:
0+1+2=3
1+2+3=6
2+3+4=9
3+4+5=12
4+5+6=15
5+6+7=18
6+7+8=21, etc...
This also proves that any three consecutive numbers can be added together to get a multiple of three. For example:
123+124+125=372.
372/3=124
You can also find the three consecutive numbers for a multiple of three by first taking the multiple of three that you want to find the three consecutive numbers for (e.g. 189):
189/3=63
63-1=62
63+1=64
So, 62+63+64=189.
Another point is that you can not get a multiple of four by adding four consecutive numbers, but you can get 5 (with 5 consecutive numbers), 7 (with 7 consecutive numbers, 9 (with 9 consecutive numbers) and etc...
This is because each of them have a median, like three, which can be multiplied by the number of consecutive numbers you have. Even one consecutive number will make a multiple of 1!