Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "2012 AMC 8 Problems/Problem 18"
m
 
(12 intermediate revisions by 9 users not shown)
Line 1: Line 1:
 +
==Problem==
 
What is the smallest positive integer that is neither prime nor square and that has no prime factor less than 50?
 
What is the smallest positive integer that is neither prime nor square and that has no prime factor less than 50?
  
Line 4: Line 5:
  
 
==Solution==
 
==Solution==
−
The problem states that the answer cannot be a perfect square or have prime factors less than <math> 50 </math>. Therefore, the answer will be the product of at least two different primes grater than <math> 50 </math>. The two smallest primes greater than <math> 50 </math> are <math> 53 </math> and <math> 59 </math>. Multiplying these two primes, we obtain the number <math> 3127 </math>, which is also the smallest number on the list of answer choices. So we are done, and the answer is <math> \boxed{\textbf{(A)}\ 3127}</math>.
+
The problem states that the answer cannot be a perfect square or have prime factors less than <math>50</math>. Therefore, the answer will be the product of at least two different primes greater than <math>50</math>. The two smallest primes greater than <math>50</math> are <math>53</math> and <math>59</math>. Multiplying these two primes, we obtain the number <math>3127</math>, which is also the smallest number on the list of answer choices.  
 +
 
 +
So we are done, and the answer is <math>\boxed{\textbf{(A)}\ 3127}</math>.
 +
 
 +
== Video Solutions ==
 +
https://youtu.be/HISL2-N5NVg?t=526
 +
 
 +
~ pi_is_3.14
 +
 
 +
https://youtu.be/qBXOgsZlCg4 ~savannahsolver
  
 
==See Also==
 
==See Also==
 
{{AMC8 box|year=2012|num-b=17|num-a=19}}
 
{{AMC8 box|year=2012|num-b=17|num-a=19}}
 +
{{MAA Notice}}

Latest revision as of 12:58, 10 November 2023

Problem

What is the smallest positive integer that is neither prime nor square and that has no prime factor less than 50?

$\textbf{(A)}\hspace{.05in}3127\qquad\textbf{(B)}\hspace{.05in}3133\qquad\textbf{(C)}\hspace{.05in}3137\qquad\textbf{(D)}\hspace{.05in}3139\qquad\textbf{(E)}\hspace{.05in}3149$

Solution

The problem states that the answer cannot be a perfect square or have prime factors less than $50$. Therefore, the answer will be the product of at least two different primes greater than $50$. The two smallest primes greater than $50$ are $53$ and $59$. Multiplying these two primes, we obtain the number $3127$, which is also the smallest number on the list of answer choices.

So we are done, and the answer is $\boxed{\textbf{(A)}\ 3127}$.

Video Solutions

https://youtu.be/HISL2-N5NVg?t=526

~ pi_is_3.14

https://youtu.be/qBXOgsZlCg4 ~savannahsolver

See Also

2012 AMC 8 (Problems • Answer Key • Resources)
Preceded by
Problem 17 		Followed by
Problem 19
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 AJHSME/AMC 8 Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2012_AMC_8_Problems/Problem_18&oldid=201892"