~~~~~~~~~~> SUMMING CONSECUTIVE NUMBERS <~~~~~~~~~~
All multiples of three can be written as the sum of three consecutive numbers. If one of the numbers you choose is one third of that number, you then need to make up the other two thirds. As you cannot have the same number twice, you simply add 1 onto one number, and take 1 away from the other number (this "splits" the numbers equally either side.)
For example, with the number 30, you would take a third of the number, which is 10. 10 + 1 = 11, and 10 - 1 = 9. Therefore 9 + 10 + 11 = 30.
Algebraically, you could use the expression "a + (a - 1) + (a + 1) = n," with 3a = n.
Multiples of 4 CANNOT be written as the sum of 4 consecutive numbers. My reason is, that if you use the method described above, you would need to quarter the number. (For my example, I am using the number 16.) 16 divided by 4 = 4. You now need to "split" the remaining 3 numbers either side of 4, and as you have 3 numbers, this is impossible.
You CANNOT write any odd numbers as the sum of 4 consecutive numbers. In a group of 4 consecutive numbers, you are going to have 2 even numbers, and 2 odd numbers, no matter where you choose to start. An even number of odd numbers always equals an even number, and an even number of even numbers always equals an even number, too. Therefore, you have 2 even numbers. As I said earlier, an even number of even numbers equals an even number, so the sum of 4 consecutive numbers will always be even.