2023 WSMO Tiebreaker Round Problems/Problem 1

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

Page

Toolbox

Search

2023 WSMO Tiebreaker Round Problems/Problem 1

Problem

Solution