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2007 iTest Problems/Problem 1: Difference between revisions

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==Problem==
==Problem==
A twin prime pair is a set of two primes <math>(p, q)</math> such that <math>q</math> is <math>2</math> greater than <math>p</math>.  What is the arithmetic mean of the two primes in the smallest twin prime pair?  
A twin prime pair is a set of two primes <math>(p, q)</math> such that <math>q</math> is <math>2</math> greater than <math>p</math>.  What is the arithmetic mean of the two primes in the smallest twin prime pair?
 
<math>\mathrm{(A)}\, 4</math>


==Solution==
==Solution==
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<math>(2,2+2)\equiv (2,4)</math>. <math>4</math> isn't a prime, so this isn't a set of twin primes.
<math>(2,2+2)\equiv (2,4)</math>. <math>4</math> isn't a prime, so this isn't a set of twin primes.


<math>(3,3+2)\equiv (3,5)</math>. <math>5</math> is a prime, so the answer is <math>\frac{3+5}{2}=4\Rightarrow \boxed{\mathrm{A}}</math>.
<math>(3,3+2)\equiv (3,5)</math>. <math>5</math> is a prime, so the answer is <math>\frac{3+5}{2}=4</math>.
===Alternate Solution===
 
Seeing as <math>A</math> is the only choice, we determine that the answer is <math>\boxed{\mathrm{A}}</math>.
==Cheap Solution==
Note that <math>A</math> is the only answer choice offered, so you must choose it.


==See Also==
==See Also==
{{iTest box|before=First question|num-a=2|year=2007}}
{{iTest box|year=2007|before=First Question|num-a=2}}


[[Category:Introductory Number Theory Problems]]
[[Category:Introductory Number Theory Problems]]

Latest revision as of 11:33, 29 January 2021

Problem

A twin prime pair is a set of two primes $(p, q)$ such that $q$ is $2$ greater than $p$. What is the arithmetic mean of the two primes in the smallest twin prime pair?

Solution

We consider the first few primes. $(2,2+2)\equiv (2,4)$. $4$ isn't a prime, so this isn't a set of twin primes.

$(3,3+2)\equiv (3,5)$. $5$ is a prime, so the answer is $\frac{3+5}{2}=4$.

Cheap Solution

Note that $A$ is the only answer choice offered, so you must choose it.

See Also

2007 iTest (Problems, Answer Key)
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Problem 2
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