2002 AMC 12B Problems/Problem 13

Problem

The sum of $18$ consecutive positive integers is a perfect square. The smallest possible value of this sum is

$\mathrm{(A)}\ 169 \qquad\mathrm{(B)}\ 225 \qquad\mathrm{(C)}\ 289 \qquad\mathrm{(D)}\ 361 \qquad\mathrm{(E)}\ 441$

Solution

Solution 1

Let the first term be $x$ and the common difference be $d.$

Thus, we know the sum of terms is $9(2x+17d).$

Because $9$ is already a perfect square, we know that $2x+17d$ must also be a perfect square.

Since we want the smallest value of the sum, we know that $25=2x+17 \Rightarrow x=4$ and $d=1.$

Thus, the answer is $9*25 = \boxed{225}.$

~coolmath2017

Solution 2

Let $a, a+1, \ldots, a + 17$ be the consecutive positive integers. Their sum, $18a + \frac{17(18)}{2} = 9(2a+17)$ , is a perfect square. Since $9$ is a perfect square, it follows that $2a + 17$ is a perfect square. The smallest possible such perfect square is $25$ when $a = 4$ , and the sum is $225 \Rightarrow \mathrm{(B)}$ .

Solution 2

Notice that all five choices given are perfect squares.

Let $a$ be the smallest number, we have $a+(a+1)+(a+2)+...+(a+17)=18a+\sum_{k=1}^{17}k=18a+153$

Subtract $153$ from each of the choices and then check its divisibility by $18$ , we have $225$ as the smallest possible sum. $\mathrm {(B)}$

~ Nafer

2002 AMC 12B (Problems • Answer Key • Resources)
Preceded by Problem 12	Followed by Problem 14
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All AMC 12 Problems and Solutions

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2002 AMC 12B Problems/Problem 13

Contents

Problem

Solution

Solution 1

Solution 2

Solution 2

See also