Difference between revisions of "2011 AMC 10B Problems/Problem 24"

Revision as of 10:56, 6 August 2019

Problem

A lattice point in an $xy$ -coordinate system is any point $(x, y)$ where both $x$ and $y$ are integers. The graph of $y = mx +2$ passes through no lattice point with $0 < x \le 100$ for all $m$ such that $\frac{1}{2} < m < a$ . What is the maximum possible value of $a$ ?

$\textbf{(A)}\ \frac{51}{101} \qquad\textbf{(B)}\ \frac{50}{99} \qquad\textbf{(C)}\ \frac{51}{100} \qquad\textbf{(D)}\ \frac{52}{101} \qquad\textbf{(E)}\ \frac{13}{25}$

Solution 1

For $y=mx+2$ to not pass through any lattice points with $0<x\leq 100$ is the same as saying that $mx\notin\mathbb Z$ for $x\in\{1,2,\dots,100\}$ , or in other words, $m$ is not expressible as a ratio of positive integers $s/t$ with $t\leq 100$ . Hence the maximum possible value of $a$ is the first real number after $1/2$ that is so expressible.

For each $d=2,\dots,100$ , the smallest multiple of $1/d$ which exceeds $1/2$ is $1,\frac23,\frac34,\frac35,\dots,\frac{50}{98},\frac{50}{99},\frac{51}{100}$ respectively, and the smallest of these is $\boxed{\textbf{(B)}\frac{50}{99}}$ .

Solution 2

We see that for the graph of $y=mx+2$ to not pass through any lattice points, the denominator of $m$ must be greater than $100$ , or else it would be canceled by some $0<x\le100$ which would make $y$ an integer. By using common denominators, we find that the order of the fractions from smallest to largest is $\text{(A), (B), (C), (D), (E)}$ . We can see that when $m=\frac{50}{99}$ , $y$ would be an integer, so therefore any fraction greater than $\frac{50}{99}$ would not work, as substituting our fraction $\frac{50}{99}$ for $m$ would produce an integer for $y$ . So now we are left with only $\frac{51}{101}$ and $\frac{50}{99}$ . But since $\frac{51}{101}=\frac{5049}{9999}$ and $\frac{50}{99}=\frac{5050}{9999}$ , we can be absolutely certain that there isn't a number between $\frac{51}{101}$ and $\frac{50}{99}$ that can reduce to a fraction whose denominator is less than or equal to $100$ . Since we are looking for the maximum value of $a$ , we take the larger of $\frac{51}{101}$ and $\frac{50}{99}$ , which is $\boxed{\textbf{(B)}\frac{50}{99}}$ .

Solution 3

We want to find the smallest $m$ such that there will be an integral solution to $y=mx+2$ with $0<x\le100$ . We first test A, but since the denominator has a $101$ , $x$ must be a nonzero multiple of $101$ , but it then will be greater than $100$ . We then test B. $y=\frac{50}{99}x+2$ yields the solution $(99,52)$ which satisfies $0<x\le100$ . Checking the answer choices, we know that the smallest possible $a$ must be $\frac{50}{99}\implies\boxed{\textbf{(B)}}$

Solution 4

Notice that for $\frac{1}{2}x=\frac{50}{100}x$ , $x=99$ is one of the integral values of $x$ such that the value of $\frac{50}{100}x$ is the closest to its next lattice point, or the next integral value of $y$ .

Revision as of 10:51, 6 August 2019 (view source) Nafer (talk \| contribs) (→‎Solution 3) ← Older edit		Revision as of 10:56, 6 August 2019 (view source) Nafer (talk \| contribs) (→‎Solution 4) Newer edit →
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	== Solution 4 ==		== Solution 4 ==
		+
		+	Notice that for <math>\frac{1}{2}x=\frac{50}{100}x</math>, <math>x=99</math> is one of the integral values of <math>x</math> such that the value of <math>\frac{50}{100}x</math> is the closest to its next lattice point, or the next integral value of <math>y</math>.

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