According to the meaning of the question, the sum of the first k numbers:1+2+3+4+...+k = (1+k) k/2,

The number k+ 1 =k+ 1

If the sum of the first k numbers is a multiple of k+ 1 number.

That is, (1+k)k/2 is divisible by (k+ 1).

Then: k/2 is a natural number (1≤k≤2002)

K is divisible by 2, and k is even.

Therefore, when k is an even number, the meaning of the problem can be satisfied, and the sum of the first k numbers is a multiple of k+ 1 number.