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 "1983 AHSME Problems/Problem 14"

(Rewrote solution for clarity)
(→‎Solution 2)
 
(2 intermediate revisions by 2 users not shown)
Line 9: Line 9:
 
\textbf{(E)}\ 9 </math>
 
\textbf{(E)}\ 9 </math>
  
−
==Solution==   
+
==Solution 1==   
  
 
First, we notice that <math>3^0</math> is congruent to <math>1 \ \text{(mod 10)}</math>, <math>3^1</math> is <math>3 \ \text{(mod 10)}</math>, <math>3^2</math> is <math>9 \ \text{(mod 10)}</math>, <math>3^3</math> is <math>7 \ \text{(mod 10)}</math>, <math>3^4</math> is <math>1 \ \text{(mod 10)}</math>, and so on. This turns out to be a cycle repeating every <math>4</math> terms, so <math>3^{1001}</math> is congruent to <math>3 \ \text{(mod 10)}</math>.
 
First, we notice that <math>3^0</math> is congruent to <math>1 \ \text{(mod 10)}</math>, <math>3^1</math> is <math>3 \ \text{(mod 10)}</math>, <math>3^2</math> is <math>9 \ \text{(mod 10)}</math>, <math>3^3</math> is <math>7 \ \text{(mod 10)}</math>, <math>3^4</math> is <math>1 \ \text{(mod 10)}</math>, and so on. This turns out to be a cycle repeating every <math>4</math> terms, so <math>3^{1001}</math> is congruent to <math>3 \ \text{(mod 10)}</math>.
  
 
The number <math>7</math> has a similar cycle, going <math>1, 7, 9, 3, 1, ...</math> Hence we have that <math>7^{1002}</math> is congruent to <math>9 \ \text{(mod 10)}</math>. Finally, <math>13^{1003}</math> is congruent to <math>3^{1003} \equiv 7 \ \text{(mod 10)}</math>. Thus the required units digit is <math>3\cdot 9\cdot 7 \equiv 9 \ \text{(mod 10)}</math>, so the answer is <math>\boxed{\textbf{(E)}\ 9}</math>.
 
The number <math>7</math> has a similar cycle, going <math>1, 7, 9, 3, 1, ...</math> Hence we have that <math>7^{1002}</math> is congruent to <math>9 \ \text{(mod 10)}</math>. Finally, <math>13^{1003}</math> is congruent to <math>3^{1003} \equiv 7 \ \text{(mod 10)}</math>. Thus the required units digit is <math>3\cdot 9\cdot 7 \equiv 9 \ \text{(mod 10)}</math>, so the answer is <math>\boxed{\textbf{(E)}\ 9}</math>.
 +
 +
==Solution 2==
 +
 +
By Euler's Totient Theorem, if <math>\gcd(a, 10) = 1</math>, then <math>a^{\phi(10)} \equiv a^4 \equiv 1\ (\mathrm{mod}\ 10)</math>.  So <math>3^{1001} \cdot 7^{1002} \cdot 13^{1003} \equiv 3^{1001} \cdot 7^{1002} \cdot 3^{1003} \equiv 3^{2004} \cdot 7^{1002} \equiv 3^0 \cdot 7^2 \equiv 1 \cdot 9 \equiv \boxed{\textbf{(E)}\ 9}\ (\mathrm{mod}\ 10)</math>.
 +
 +
-j314andrews
 +
 +
==See Also==
 +
{{AHSME box|year=1983|num-b=13|num-a=15}}
 +
 +
{{MAA Notice}}

Latest revision as of 15:10, 3 July 2025

Problem

The units digit of $3^{1001} 7^{1002} 13^{1003}$ is

$\textbf{(A)}\ 1\qquad \textbf{(B)}\ 3\qquad \textbf{(C)}\ 5\qquad \textbf{(D)}\ 7\qquad \textbf{(E)}\ 9$

Solution 1

First, we notice that $3^0$ is congruent to $1 \ \text{(mod 10)}$, $3^1$ is $3 \ \text{(mod 10)}$, $3^2$ is $9 \ \text{(mod 10)}$, $3^3$ is $7 \ \text{(mod 10)}$, $3^4$ is $1 \ \text{(mod 10)}$, and so on. This turns out to be a cycle repeating every $4$ terms, so $3^{1001}$ is congruent to $3 \ \text{(mod 10)}$.

The number $7$ has a similar cycle, going $1, 7, 9, 3, 1, ...$ Hence we have that $7^{1002}$ is congruent to $9 \ \text{(mod 10)}$. Finally, $13^{1003}$ is congruent to $3^{1003} \equiv 7 \ \text{(mod 10)}$. Thus the required units digit is $3\cdot 9\cdot 7 \equiv 9 \ \text{(mod 10)}$, so the answer is $\boxed{\textbf{(E)}\ 9}$.

Solution 2

By Euler's Totient Theorem, if $\gcd(a, 10) = 1$, then $a^{\phi(10)} \equiv a^4 \equiv 1\ (\mathrm{mod}\ 10)$. So $3^{1001} \cdot 7^{1002} \cdot 13^{1003} \equiv 3^{1001} \cdot 7^{1002} \cdot 3^{1003} \equiv 3^{2004} \cdot 7^{1002} \equiv 3^0 \cdot 7^2 \equiv 1 \cdot 9 \equiv \boxed{\textbf{(E)}\ 9}\ (\mathrm{mod}\ 10)$.

-j314andrews

See Also

1983 AHSME (Problems • Answer Key • Resources)
Preceded by
Problem 13 		Followed by
Problem 15
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
All AHSME Problems and Solutions


These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. AMC Logo.png

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=1983_AHSME_Problems/Problem_14&oldid=253322"