2019 AMC 10A Problems/Problem 9
Contents
Problem
What is the greatest three-digit positive integer for which the sum of the first
positive integers is
a divisor of the product of the first
positive integers?
Solutions
Solution 1
The sum of the first positive integers is
, and we want this not to be a divisor of
(the product of the first
positive integers). Notice that if and only if
were composite, all of its factors would be less than or equal to
, which means they would be able to cancel with the factors in
. Thus, the sum of
positive integers would be a divisor of
when
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,
must instead be prime. The greatest three-digit integer that is prime is
, so we subtract
to get
.
Solution 2
As in Solution 1, we deduce that 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
. Choices
,
, and
don't work because
is even, and all even numbers are divisible by two, which makes choices
,
, and
composite and not prime. Choice
also does not work since
is divisible by
, which means it's a composite number and not prime. Thus, the correct answer must be
.
Video Solutions
Video Solution 1
~savannahsolver
Video Solution 2 by OmegaLearn
https://youtu.be/FDgcLW4frg8?t=33
~ pi_is_3.14
Video Solution 3
Education, the Study of Everything
See Also
2019 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 8 |
Followed by Problem 10 | |
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 | ||
All AMC 10 Problems and Solutions |
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