Art of Problem Solving
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

2007 iTest Problems/Problem 11: Difference between revisions

Timneh (talk | contribs)
2,460 edits
m Created page with "== Problem 11 == Consider the "tower of power" <math>2^{2^{2^{\cdot^{\cdot^\cdot^{2}}}}}</math>, where there are 2007 twos including the base. What is the last (units digit) of t..."
 
Rockmanex3 (talk | contribs)
editor
1,245 edits
 
(One intermediate revision by the same user not shown)
Line 1: Line 1:
== Problem 11 ==
== Problem 11 ==
Consider the "tower of power" <math>2^{2^{2^{\cdot^{\cdot^\cdot^{2}}}}}</math>, where there are 2007 twos including the base. What is the last (units digit) of this number?
Consider the "tower of power" <math>2^{2^{2^{\cdot^{\cdot^{\cdot^{2}}}}}}</math>, where there are 2007 twos including the base. What is the last (units digit) of this number?


<math>\text{(A) }0\qquad
<math>\text{(A) }0\qquad
Line 15: Line 15:


== Solution ==
== Solution ==
Note that <math>2^1 = 2</math>, <math>2^2 = 4</math>, <math>2^3 = 8</math>, <math>2^4 = 16</math>, and <math>2^5 = 32</math>.  The units digit of <math>2^n</math> cycle every time <math>n</math> is increased by <math>4</math>.
Since <math>2^{2^{2^{\cdot^{\cdot^{\cdot^{2}}}}}}</math> with <math>2006</math> twos is a multiple of four, the units digit is <math>\boxed{\textbf{(G) }6}</math>.
==See Also==
{{iTest box|year=2007|num-b=10|num-a=12}}
[[Category:Introductory Number Theory Problems]]

Latest revision as of 03:31, 10 June 2018

Problem 11

Consider the "tower of power" $2^{2^{2^{\cdot^{\cdot^{\cdot^{2}}}}}}$, where there are 2007 twos including the base. What is the last (units digit) of this number?

$\text{(A) }0\qquad \text{(B) }1\qquad \text{(C) }2\qquad \text{(D) }3\qquad \text{(E) }4\qquad \text{(F) }5\qquad \text{(G) }6\qquad \text{(H) }7\qquad \text{(I) }8\qquad \text{(J) }9\qquad \text{(K) }2007\qquad$

Solution

Note that $2^1 = 2$, $2^2 = 4$, $2^3 = 8$, $2^4 = 16$, and $2^5 = 32$. The units digit of $2^n$ cycle every time $n$ is increased by $4$.

Since $2^{2^{2^{\cdot^{\cdot^{\cdot^{2}}}}}}$ with $2006$ twos is a multiple of four, the units digit is $\boxed{\textbf{(G) }6}$.

See Also

2007 iTest (Problems, Answer Key)
Preceded by:
Problem 10 		Followed by:
Problem 12
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 • 31 • 32 • 33 • 34 • 35 • 36 • 37 • 38 • 39 • 40 • 41 • 42 • 43 • 44 • 45 • 46 • 47 • 48 • 49 • 50 • 51 • 52 • 53 • 54 • 55 • 56 • 57 • 58 • 59 • 60 • TB1 • TB2 • TB3 • TB4
Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2007_iTest_Problems/Problem_11&oldid=95033"