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 n th positive integer appears n times. The remainder when the 1993 rd term is divided by 5 is

1993 AMC 12 Problem 16

Attempt Problem 16 of the 1993 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1993 AMC 12 solutions, or check the answer key.

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16.

Consider the non-decreasing sequence of positive integers

\begin{aligned} &1,2,2,3,3,3,4,4,4,4,\\ &5,5,5,5,5,\ldots \end{aligned}

in which the

n

th positive integer appears

n

times. The remainder when the

1993

rd term is divided by

5

0

1

2

3

4

The final occurrence of

n

is in position

\frac{n(n+1)}2

Compare

1993

with the triangular numbers for

62

and

63

The last

62

occurs in position

\frac{62\cdot63}{2}=1953,

while the last

63

occurs in position

\frac{63\cdot64}{2}=2016.

Thus the

1993

rd term is

63,

whose remainder modulo

5

3.

Thus the correct answer is D.