Difference between revisions of "1967 AHSME Problems/Problem 24"

(Solution)
 
Line 11: Line 11:
 
We have <math>y = \frac{501 - 3x}{5}</math>.  Thus, <math>501 - 3x</math> must be a positive multiple of <math>5</math>.  If <math>x = 2</math>, we find our first positive multiple of <math>5</math>.  From there, we note that <math>x = 2 + 5k</math> will always return a multiple of <math>5</math> for <math>501 - 3x</math>.  Our first solution happens at <math>k=0</math>.   
 
We have <math>y = \frac{501 - 3x}{5}</math>.  Thus, <math>501 - 3x</math> must be a positive multiple of <math>5</math>.  If <math>x = 2</math>, we find our first positive multiple of <math>5</math>.  From there, we note that <math>x = 2 + 5k</math> will always return a multiple of <math>5</math> for <math>501 - 3x</math>.  Our first solution happens at <math>k=0</math>.   
  
We now want to find the smallest multiple of <math>5</math> that will work.  If <math>x = 2 + 5k</math>, then we have <math>501 - 3x = 501 - 3(2 + 5k)</math>, or <math>495 - 15k</math>.  When <math>k = 32</math>, the expression is equal to 15<math>, and when </math>k = 33<math>, the expression is equal to </math>0<math>, which will no longer work.
 
  
Thus, all integers from </math>k = 0<math> to </math>k = 32<math> will generate an </math>x = 2 + 5k<math> that will be a positive integer, and which will in turn generate a </math>y<math> that is also a positive integer.  So, the answer is </math>\fbox{A}$.
+
We now want to find the smallest multiple of <math>5</math> that will work.  If <math>x = 2 + 5k</math>, then we have <math>501 - 3x = 501 - 3(2 + 5k)</math>, or <math>495 - 15k</math>.  When <math>k = 32</math>, the expression is equal to <math>15</math>, and when <math>k = 33</math>, the expression is equal to <math>0</math>, which will no longer work.
 +
 
 +
 
 +
Thus, all integers from <math>k = 0</math> to <math>k = 32</math> will generate an <math>x = 2 + 5k</math> that will be a positive integer, and which will in turn generate a <math>y</math> that is also a positive integer.  So, the answer is <math>\fbox{A}</math>.
  
 
== See also ==
 
== See also ==

Latest revision as of 23:58, 12 July 2019

Problem

The number of solution-pairs in the positive integers of the equation $3x+5y=501$ is:

$\textbf{(A)}\ 33\qquad \textbf{(B)}\ 34\qquad \textbf{(C)}\ 35\qquad \textbf{(D)}\ 100\qquad \textbf{(E)}\ \text{none of these}$

Solution

We have $y = \frac{501 - 3x}{5}$. Thus, $501 - 3x$ must be a positive multiple of $5$. If $x = 2$, we find our first positive multiple of $5$. From there, we note that $x = 2 + 5k$ will always return a multiple of $5$ for $501 - 3x$. Our first solution happens at $k=0$.


We now want to find the smallest multiple of $5$ that will work. If $x = 2 + 5k$, then we have $501 - 3x = 501 - 3(2 + 5k)$, or $495 - 15k$. When $k = 32$, the expression is equal to $15$, and when $k = 33$, the expression is equal to $0$, which will no longer work.


Thus, all integers from $k = 0$ to $k = 32$ will generate an $x = 2 + 5k$ that will be a positive integer, and which will in turn generate a $y$ that is also a positive integer. So, the answer is $\fbox{A}$.

See also

1967 AHSME (ProblemsAnswer KeyResources)
Preceded by
Problem 23
Followed by
Problem 25
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

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

Invalid username
Login to AoPS