Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "2019 Mock AMC 10B Problems/Problem 19"
Line 7: Line 7:
 
== Solution ==  
 
== Solution ==  
  
−
<math>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) = (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)</math>. By simple mod checking, we find that <math>3^{1008} + 1 \equiv 3^{504} + 1 \equiv e^{252} + 1 \equiv 3^{126} + 1 \equiv 3^{63} - 1 \equiv 2</math> <math>\text{mod}</math> <math>4</math>, and <math>3^{63} + 1 \equiv 4</math> <math>\text{mod}</math> <math>8</math>. Therefore, the smallest powers of <math>2</math> that divide each of these numbers are <math>2, 2, 2, 2, 2, and 4</math>, respectively. The smallest power of <math>2</math> that divides <math>3^{2016} - 1</math> is thus <math>2^5 \cdot 4 = \boxed{\text{E} 256}</math>.
+
<math>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) = (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)</math>. By simple mod checking, we find that <math>3^{1008} + 1 \equiv 3^{504} + 1 \equiv e^{252} + 1 \equiv 3^{126} + 1 \equiv 3^{63} - 1 \equiv 2</math> <math>\text{mod}</math> <math>4</math>, and <math>3^{63} + 1 \equiv 4</math> <math>\text{mod}</math> <math>8</math>. Therefore, the smallest powers of <math>2</math> that divide each of these numbers are <math>2, 2, 2, 2, 2, and 4</math>, respectively. The smallest power of <math>2</math> that divides <math>3^{2016} - 1</math> is thus <math>2^5 \cdot 4 = \boxed{\text{E} 256}</math>.

Revision as of 20:46, 2 November 2019

Problem

What is the largest power of $2$ that divides $3^{2016}-1$?

$\textbf{(A)}\ 16 \qquad\textbf{(B)}\ 32 \qquad\textbf{(C)}\ 64 \qquad\textbf{(D)}\ 128 \qquad\textbf{(E)}\ 256$

Solution

$3^{2016} - 1 = (3^{1008} - 1)(3^{1008} + 1) = (3^{504} - 1)(3^{504} + 1)(3^{1008} + 1) = (3^{252} - 1)(3^{252} + 1)(3^{504} + 1)(3^{1008} + 1) = (3^{126} - 1)(3^{126} + 1)(3^{252} + 1)(3^{504} + 1)(3^{1008} + 1) = (3^{63} - 1)(3^{63} + 1)(3^{126} + 1)(3^{252} + 1)(3^{504} + 1)(3^{1008} + 1)$. By simple mod checking, we find that $3^{1008} + 1 \equiv 3^{504} + 1 \equiv e^{252} + 1 \equiv 3^{126} + 1 \equiv 3^{63} - 1 \equiv 2$ $\text{mod}$ $4$, and $3^{63} + 1 \equiv 4$ $\text{mod}$ $8$. Therefore, the smallest powers of $2$ that divide each of these numbers are $2, 2, 2, 2, 2, and 4$, respectively. The smallest power of $2$ that divides $3^{2016} - 1$ is thus $2^5 \cdot 4 = \boxed{\text{E} 256}$.

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2019_Mock_AMC_10B_Problems/Problem_19&oldid=110784"