2019 Mock AMC 10B Problems/Problem 19
Problem
What is the largest power of that divides ?
Solution
$$ (Error compiling LaTeX. Unknown error_msg)3^{2016} - 1 = (3^{1008} - 1)(3^{1008} + 1) = (3^{504} - 1)(3^{504} + 1)(3^{1008} + 1) = (3^{252} - 1)(3^{252} + 1)(3^{504} + 1)(3^{1008} + 1)$$ (Error compiling LaTeX. Unknown error_msg)= (3^{126} - 1)(3^{126} + 1)(3^{252} + 1)(3^{504} + 1)(3^{1008} + 1) = (3^{63} - 1)(3^{63} + 1)(3^{126} + 1)(3^{252} + 1)(3^{504} + 1)(3^{1008} + 1)$$ (Error compiling LaTeX. Unknown error_msg). By simple mod checking, we find that , and . Therefore, the smallest powers of that divide each of these numbers are , respectively. The smallest power of that divides is thus .