Question

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 $N?$

$6$

$7$

$8$

$9$

$10$

Answer: D

Concepts:triangular number factoring digits

Difficulty rating: 1070

Solution:

Recall that the sum of the first $N$ numbers is $\dfrac{N(N + 1)}{2}.$

We want to find $N$ such that $\dfrac{N(N + 1)}{2} = 2016.$ Cross-multiplying and simplifying gives us $N^2 + N - 4032 = 0.$ Factoring gives us $(N - 63)(N + 64) = 0.$ We want the positive value so $N = 63.$ Adding together the digits gives us $9.$

Thus, the correct answer is D .

2016 AMC 10A Problem 9

Answer: D

Difficulty rating: 1070

Solution:

Problem 9 in Other Years