Difference between revisions of "2019 AMC 10A Problems/Problem 9"

Revision as of 03:17, 16 January 2023

Problem

What is the greatest three-digit positive integer $n$ for which the sum of the first $n$ positive integers is $\underline{not}$ a divisor of the product of the first $n$ positive integers?

$\textbf{(A) } 995 \qquad\textbf{(B) } 996 \qquad\textbf{(C) } 997 \qquad\textbf{(D) } 998 \qquad\textbf{(E) } 999$

Solutions

Solution 1

The sum of the first $n$ positive integers is $\frac{(n)(n+1)}{2}$ , and we want this not to be a divisor of $n!$ (the product of the first $n$ positive integers). Notice that if and only if $n+1$ were composite, all of its factors would be less than or equal to $n$ , which means they would be able to cancel with the factors in $n!$ . Thus, the sum of $n$ positive integers would be a divisor of $n!$ when $n+1$ is composite. (Note: This is true for all positive integers except for 1 because 2 is not a divisor/factor of 1.) Hence in this case, $n+1$ must instead be prime. The greatest three-digit integer that is prime is $997$ , so we subtract $1$ to get $n=\boxed{\textbf{(B) } 996}$ .

Solution 2

As in Solution 1, we deduce that $n+1$ must be prime. If we can't immediately recall what the greatest three-digit prime is, we can instead use this result to eliminate answer choices as possible values of $n$ . Choices $A$ , $C$ , and $E$ don't work because $n+1$ is even, and all even numbers are divisible by two, which makes choices $A$ , $C$ , and $E$ composite and not prime. Choice $D$ also does not work since $999$ is divisible by $9$ , which means it's a composite number and not prime. Thus, the correct answer must be $\boxed{\textbf{(B) } 996}$ .

Video Solutions

Video Solution 1

https://youtu.be/2vucE8HTiuU

~savannahsolver

Video Solution 2 by OmegaLearn

https://youtu.be/FDgcLW4frg8?t=33

~ pi_is_3.14

Video Solution 3

https://youtu.be/ikRv_0kNc2w

Education, the Study of Everything

@@ Line 16: / Line 16: @@
 ~savannahsolver
-===Video Solution by OmegaLearn===
+===Video Solution 2 by OmegaLearn===
 https://youtu.be/FDgcLW4frg8?t=33

2019 AMC 10A (Problems • Answer Key • Resources)
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