Please prove that for any n natural numbers, one or the sum of several numbers must be a multiple of n

Please prove that for any n natural numbers, one or the sum of several numbers must be a multiple of n

If one of N numbers is a multiple of N, it is obviously true
If not, the remainder of N divided by N is 1 ,n-1
There must be some two numbers with the same remainder. If the remainder is the sum of 1 and n-1, it can divide n exactly
So for any n natural numbers, one or the sum of several numbers must be a multiple of n