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 "1964 AHSME Problems/Problem 3"

(Solution $1$=)
(Solution $1$=)
Line 14: Line 14:
  
 
*We can solve this problem by elemetary modular arthmetic,
 
*We can solve this problem by elemetary modular arthmetic,
−
<math>x</math> \equiv. <math>v</math> (<math>mod y</math>) <math>=></math> <math>x</math> + <math>2uy</math> \equiv. <math>v</math> (<math>mod y</math>).
+
<math>x</math> \equiv. <math>v</math> (<math>mod y</math>) <math>=></math> <math>x</math> + <math>2uy</math> \equiv <math>v</math> (<math>mod y</math>).
  
−
% Solution by GEOMETRY-WIZARD<math>
+
<math>Solution by GEOMETRY-WIZARD</math>
  
 
==Solution 2==
 
==Solution 2==
  
−
By the definition of quotient and remainder, problem states that </math>x = uy + v<math>.
+
By the definition of quotient and remainder, problem states that <math>x = uy + v</math>.
  
−
The problem asks to find the remainder of </math>x + 2uy<math> when divided by </math>y<math>.  Since </math>2uy<math> is divisible by </math>y<math>, adding it to </math>x<math> will not change the remainder.  Therefore, the answer is </math>\boxed{\textbf{(D)}}<math>.
+
The problem asks to find the remainder of <math>x + 2uy</math> when divided by <math>y</math>.  Since <math>2uy</math> is divisible by <math>y</math>, adding it to <math>x</math> will not change the remainder.  Therefore, the answer is <math>\boxed{\textbf{(D)}}</math>.
  
 
==Solution 3==
 
==Solution 3==
−
If the statement is true for all values of </math>(x, y, u, v)<math>, then it must be true for a specific set of </math>(x, y, u, v)<math>.
+
If the statement is true for all values of <math>(x, y, u, v)</math>, then it must be true for a specific set of <math>(x, y, u, v)</math>.
  
−
If you let </math>x=43<math> and </math>y = 8<math>, then the quotient is </math>u = 5<math> and the remainder is </math>v = 3<math>.  The problem asks what the remainder is when you divide </math>x + 2uy = 43 + 2 \cdot 5 \cdot 8 = 123<math> by </math>8<math>.  In this case, the remainder is </math>3<math>.
+
If you let <math>x=43</math> and <math>y = 8</math>, then the quotient is <math>u = 5</math> and the remainder is <math>v = 3</math>.  The problem asks what the remainder is when you divide <math>x + 2uy = 43 + 2 \cdot 5 \cdot 8 = 123</math> by <math>8</math>.  In this case, the remainder is <math>3</math>.
  
−
When you plug in </math>u=5<math> and </math>v = 3<math> into the answer choices, they become </math>0, 5, 10, 3, 6<math>, respectively.  Therefore, the answer is </math>\boxed{\textbf{(D)}}$.
+
When you plug in <math>u=5</math> and <math>v = 3</math> into the answer choices, they become <math>0, 5, 10, 3, 6</math>, respectively.  Therefore, the answer is <math>\boxed{\textbf{(D)}}</math>.
  
 
==See Also==
 
==See Also==

Revision as of 05:40, 31 December 2023

Problem

When a positive integer $x$ is divided by a positive integer $y$, the quotient is $u$ and the remainder is $v$, where $u$ and $v$ are integers. What is the remainder when $x+2uy$ is divided by $y$?

$\textbf{(A)}\ 0 \qquad \textbf{(B)}\ 2u \qquad \textbf{(C)}\ 3u \qquad \textbf{(D)}\ v \qquad \textbf{(E)}\ 2v$


Solution 1

  • We can solve this problem by elemetary modular arthmetic,

$x$ \equiv. $v$ ($mod y$) $=>$ $x$ + $2uy$ \equiv $v$ ($mod y$).

$Solution by GEOMETRY-WIZARD$

Solution 2

By the definition of quotient and remainder, problem states that $x = uy + v$.

The problem asks to find the remainder of $x + 2uy$ when divided by $y$. Since $2uy$ is divisible by $y$, adding it to $x$ will not change the remainder. Therefore, the answer is $\boxed{\textbf{(D)}}$.

Solution 3

If the statement is true for all values of $(x, y, u, v)$, then it must be true for a specific set of $(x, y, u, v)$.

If you let $x=43$ and $y = 8$, then the quotient is $u = 5$ and the remainder is $v = 3$. The problem asks what the remainder is when you divide $x + 2uy = 43 + 2 \cdot 5 \cdot 8 = 123$ by $8$. In this case, the remainder is $3$.

When you plug in $u=5$ and $v = 3$ into the answer choices, they become $0, 5, 10, 3, 6$, respectively. Therefore, the answer is $\boxed{\textbf{(D)}}$.

See Also

1964 AHSC (Problems • Answer Key • Resources)
Preceded by
Problem 2 		Followed by
Problem 4
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
All AHSME Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=1964_AHSME_Problems/Problem_3&oldid=209113"