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Difference between revisions of "2012 AMC 8 Problems/Problem 18"

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==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?
  
 
<math> \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 </math>
 
<math> \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 </math>
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==Solution==
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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.
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So we are done, and the answer is <math>\boxed{\textbf{(A)}\ 3127}</math>.
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== Video Solutions ==
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https://youtu.be/HISL2-N5NVg?t=526
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~ pi_is_3.14
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https://youtu.be/qBXOgsZlCg4 ~savannahsolver
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==See Also==
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{{AMC8 box|year=2012|num-b=17|num-a=19}}
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{{MAA Notice}}

Latest revision as of 13:58, 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

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