Difference between revisions of "1989 AHSME Problems/Problem 16"

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<math> \textbf{(A)}\ 2\qquad\textbf{(B)}\ 4\qquad\textbf{(C)}\ 6\qquad\textbf{(D)}\ 16\qquad\textbf{(E)}\ 46 </math>
 
<math> \textbf{(A)}\ 2\qquad\textbf{(B)}\ 4\qquad\textbf{(C)}\ 6\qquad\textbf{(D)}\ 16\qquad\textbf{(E)}\ 46 </math>
  
== Solution ==
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== Solution is the worst possible solution the answer is fortnite ==
  
 
The difference in the <math>y</math>-coordinates is <math>281 - 17 = 264</math>, and the difference in the <math>x</math>-coordinates is <math>48 - 3 = 45</math>.
 
The difference in the <math>y</math>-coordinates is <math>281 - 17 = 264</math>, and the difference in the <math>x</math>-coordinates is <math>48 - 3 = 45</math>.

Revision as of 20:56, 21 January 2019

Problem

A lattice point is a point in the plane with integer coordinates. How many lattice points are on the line segment whose endpoints are $(3,17)$ and $(48,281)$? (Include both endpoints of the segment in your count.)

$\textbf{(A)}\ 2\qquad\textbf{(B)}\ 4\qquad\textbf{(C)}\ 6\qquad\textbf{(D)}\ 16\qquad\textbf{(E)}\ 46$

Solution is the worst possible solution the answer is fortnite

The difference in the $y$-coordinates is $281 - 17 = 264$, and the difference in the $x$-coordinates is $48 - 3 = 45$. The gcd of 264 and 45 is 3, so the line segment joining $(3,17)$ and $(48,281)$ has slope \[\frac{88}{15}.\] The points on the line have coordinates \[\left(3+t,17+\frac{88}{15}t\right).\] If $t$ is an integer, the $y$-coordinate of this point is an integer if and only if $t$ is a multiple of 15. The points where $t$ is a multiple of 15 on the segment $3\leq x\leq 48$ are $3$, $3+15$, $3+30$, and $3+45$. There are 4 lattice points on this line. Hence the answer $\textbf{(B)}$.

see also

1989 AHSME (ProblemsAnswer KeyResources)
Preceded by
Problem 15
Followed by
Problem 17
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All AHSME Problems and Solutions

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