The sum of n different positive integers is less than 100 and we need to find the greatest possible value of n
So, the n different positive integers should be least
And we know that Sum of first n positive integers = \(\frac{n*(n+1)}{2}\)

=> \(\frac{n*(n+1)}{2}\) < 100
=> n*(n+1) < 200

Lets start with n = 12 we get
12*(12+1)= 12*13 = 156 < 200 , So possible

Lets go higher then, n = 13
13*14 = 182 < 200, so possible

We need to check with 14 now
14*15 = 210 < 200 Not possible

Hence answer will be D
Hope it helps!