Problem:
Consider the non-decreasing sequence of positive integers
1,2,2,3,3,3,4,4,4,4,5,5,5,5,5,..
in which the nth positive integer appears n times. The remainder when the 1993rd term is divided by 5 is
Answer Choices:
A. 0
B. 1
C. 2
D. 3
E. 4
Solution:
The last occurrence of the nth positive integer is in position number
1+2+⋯+n=2n(n+1)​
To approximate our n, consider 2n2​=2000. Since 262(63)​=1953 and 263(64)​=1953+63=2016, it follows that the 1993rd term is 63, which leaves a remainder of 3 when divided by 5.