Question 1079492
.
What is the sum of the two smallest distinct prime factors of 2^{27} + 3^{27}?'
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<U>Answer</U>. This sum is 5 + 7 = 12.


<U>Solution</U>


<pre>
1.  It is clear that 2 is not a factor of  {{{2^27 + 3^27}}}.



2.  It is clear that 3 is not a factor of  {{{2^27 + 3^27}}}.



3.  {{{2^27 + 3^27}}}  has the factor (2 + 3) = 5  since  {{{2^27 + 3^27}}}  has this well known decomposition in the product of these factors:


    {{{2^27 + 3^27}}} = {{{(2 + 3)*(2^26 - 2^25*3^1 + 2^24*3^2 - 2^23*3^4 + ellipsis - 2^1*3^25 + 3^26)}}}


    Thus the first smallest prime factor of the sum  {{{2^27 + 3^27}}}  is 5.



4.  {{{2^27 + 3^27}}}  has the factor {{{2^3 + 3^3)}}} = 35  since  {{{2^27 + 3^27}}}  can be factored in this way:
   

    {{{2^27 + 3^27}}}  = {{{(2^3)^9 + (3^3)^9}}} = {{{8^9 + 27^9}}} = {{{(8+27)*(8^8 - 8^7*27^1 + 8^6*27^2 - ellipsis - 8^1*27^7 + 27^8)}}}


    and (8+27) = 35 is multiple of 7,  so the second smallest prime divisor of this sum is 7 (next after 5).


5.  Therefore, 5 + 7 = 12 is the answer.
</pre>

Solved.



On factoring binomials {{{x^n-a^n}}} and {{{x^n+a^n}}} see the lessons

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Polynomials-and-rational-expressions/Divisibility-of-the-Binomial-%28x%5En-a%5En%29-by-%28x-a%29.lesson>Factoring the binomials &nbsp;&nbsp;{{{x^n-a^n}}}</A> 

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Polynomials-and-rational-expressions/Factoring-the-binomials-x%5En+a%5En-for-odd-degrees.lesson>Factoring the binomials &nbsp;&nbsp;{{{x^n+a^n}}} for odd degrees</A>

in this site.



Also, &nbsp;you have this free of charge online textbook in ALGEBRA-I in this site

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-I - YOUR ONLINE TEXTBOOK</A>.


The referred lessons are the part of this online textbook under the topic 
"<U>Factoring binomials {{{x^n-a^n}}} and {{{x^n+a^n}}}</U>".