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Difference between revisions of "2023 WSMO Tiebreaker Round Problems/Problem 1"

(Created page with "==Problem== Find the number of factors of <math>24 ^ {6} - 20 ^ {6}.</math> ==Solution==")
 
 
Line 4: Line 4:
  
 
==Solution==
 
==Solution==
 +
Note that
 +
<cmath>\begin{align*}
 +
24^6-20^6 &= (24^3-20^3)(24^3+20^3)\\
 +
&= (24-20)(24^2+24\cdot20+20^2)(24+20)(24^2-24\cdot20+20^2)\\
 +
&= (4)(1456)(44)(496)\\
 +
&= 2^2\cdot(2^4\cdot31)(2^2\cdot11)(2^4\cdot7\cdot13)\\
 +
&= 2^{12}\cdot7\cdot11\cdot13\cdot31,
 +
\end{align*}</cmath>
 +
which has <cmath>(12+1)(1+1)(1+1)(1+1)(1+1)=(13)(2)(2)(2)(2)=\boxed{208}.</cmath>
 +
 +
~pinkpig

Latest revision as of 10:34, 15 September 2025

Problem

Find the number of factors of $24 ^ {6} - 20 ^ {6}.$

Solution

Note that \begin{align*} 24^6-20^6 &= (24^3-20^3)(24^3+20^3)\\ &= (24-20)(24^2+24\cdot20+20^2)(24+20)(24^2-24\cdot20+20^2)\\ &= (4)(1456)(44)(496)\\ &= 2^2\cdot(2^4\cdot31)(2^2\cdot11)(2^4\cdot7\cdot13)\\ &= 2^{12}\cdot7\cdot11\cdot13\cdot31, \end{align*} which has \[(12+1)(1+1)(1+1)(1+1)(1+1)=(13)(2)(2)(2)(2)=\boxed{208}.\]

~pinkpig

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