2016 AMC 12A Problem 6

Below is the professionally curated solution for Problem 6 of the 2016 AMC 12A, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2016 AMC 12A solutions, or check the answer key.

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Concepts:triangular number estimation

Difficulty rating: 1270

A triangular array of $2016$ coins has $1$ coin in the first row, $2$ coins in the second row, $3$ coins in the third row, and so on up to $N$ coins in the $N$ th row. What is the sum of the digits of $N?$

$6$

$7$

$8$

$9$

$10$

Solution:

The total number of coins is $1+2+\cdots+N=\dfrac{N(N+1)}{2}=2016,$ so $N(N+1)=4032.$ Since $63\cdot 64=4032,$ we have $N=63,$ and the sum of its digits is $6+3=9.$

Thus, the correct answer is D.

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