# 1956 AHSME Problems/Problem 36

## Problem 36

If the sum $1 + 2 + 3 + \cdots + K$ is a perfect square $N^2$ and if $N$ is less than $100$, then the possible values for $K$ are:

$\textbf{(A)}\ \text{only }1\qquad \textbf{(B)}\ 1\text{ and }8\qquad \textbf{(C)}\ \text{only }8\qquad \textbf{(D)}\ 8\text{ and }49\qquad \textbf{(E)}\ 1,8,\text{ and }49$

## Solution

We can plug in 1, 8, and 49 to see which works.

$1 = 1 = 1^2$

$1 + 2 + 3 + \cdots + 8 = \frac{8 \cdot 9}{2} = 36 = 6^2$

$1 + 2 + 3 + \cdots + 49 = \frac{49 \cdot 50}{2} = 1225 = 45^2$

All of these values produce a perfect square for $1 + 2 + \cdots + K,$ so the answer is $\boxed{\textbf{(E)}}.$

-coolmath34

## See Also

 1956 AHSC (Problems • Answer Key • Resources) Preceded byProblem 35 Followed byProblem 37 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 • 31 • 32 • 33 • 34 • 35 • 36 • 37 • 38 • 39 • 40 • 41 • 42 • 43 • 44 • 45 • 46 • 47 • 48 • 49 • 50 All AHSME Problems and Solutions

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