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Yash Agarwal

#3:

For the first part of the problem, if you look at the formula, it is (n(n+1))/2. So, if n is even, then n+1 is odd and vice versa. If n is even, then the formula is equal to:

(n+1) * n/2. This is a factor of n+1, not n. So, that means n has to be odd to make this work. So, the answer is just all the odd numbers until 100. There are **50 odd numbers.**

For the second part of this problem, the most obvious possible answer is 1. 1 (mod 1) = 0. The next possible answer is 5, since n+1 is 6 and 6 divides 6 evenly. Then, the next possible number is 9, where 10*9 (mod 2*3(6)) is 0 because 10 (mod 2) = 0 and 9 (mod 3) equals 0. So, the pattern is 1,5,9,13… until 97. That is ** 25 numbers. **

**This response is written by Yash Agarwal. **