Difference between revisions of "2014 AIME I Problems/Problem 3"

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Problem 3

Find the number of rational numbers $r,$ $0<r<1,$ such that when $r$ is written as a fraction in lowest terms, the numerator and the denominator have a sum of 1000.

Solution 1

We have that the set of these rational numbers is from $\dfrac{1}{999}$ to $\dfrac{499}{501}$ where each each element $\dfrac{n}{m}$ has $n+m =1000$ and $\dfrac{n}{m}$ is irreducible.

We note that $\dfrac{n}{m} =\dfrac{1000-m}{m}=\dfrac{1000}{m}-1$ . Hence, $\dfrac{n}{m}$ is irreducible if $\dfrac{1000}{m}$ is irreducible, and $\dfrac{1000}{m}$ is irreducible if $m$ is not divisible by 2 or 5. Thus, the answer to the question is the number of integers between 999 and 501 inclusive that are not divisible by 2 or 5.

We note there are 499 numbers between 501 and 999, and

249 numbers are divisible by 2
99 numbers are divisible by 5
49 numbers are divisible by 10

Using the Principle of Inclusion and Exclusion, we get that there are $499-249-99+49=200$ numbers between $501$ and $999$ are not divisible by either $2$ or $5$ , so our answer is $\boxed{200}$ .

Euler's Totient Function can also be used to arrive at 400 numbers relatively prime to 1000, meaning 200 possible fractions satisfying the necessary conditions.

Solution 2

If the initial manipulation is not obvious, instead ,consider the euclidean algorithm. Instead of using $\frac{n}{m}$ as the fraction to use the euclidean algorithm on, rewrite this as $\frac{500-x}{500+x} gcd(500+x,500-x)=gcd((500+x)+(500-x),500-x)=gcd(1000,500-x)$ . Thus, we want $gcd(1000,500-x)=1$ . You can either proceed as solution 1, or consider that no even numbers work, limiting us to $250$ choices of numbers and restricting $x$ to be odd. If $x$ is odd, $500-x$ is odd, so the only possible common factors $1000$ and $500-x$ can share are multiples of $5$ . Thus, we want to avoid these. There are $50$ multiples of $5$ less than $500$ , so the answer is $250-50=\boxed{200}$ .

Solution Numbers

Say $r=\frac{d}{1000-d}$ ; then $1<=d<=499$ . If this fraction is reducible, then the modulus of some number for $d$ is the same as the modulus for $1000-d$ . Since $1000=2^3*5^3$ , that modulus can only be $2$ or $5$ . This implies that if $d|2$ or $d|5$ , the fraction is reducible. There are 249 cases where $d|2$ , 99 where $d|5$ , and 49 where $d|(2*5=10)$ , so by PIE, the number of fails is 299, so our answer is $\boxed{200}$ .

@@ Line 20: / Line 20: @@
 ==Solution Numbers==
-Say <math>r=\frac{d}{1000-d}</math>; then <math>1<=d<=499</math>. If this fraction is reducible, then the modulus of some number for <math>d</math> is the same as the modulus for <math>1000-d</math>. Since <math>1000=2^3*5*3</math>, that modulus can only be <math>2</math> or <math>5</math>. This implies that if <math>d|2</math> or <math>d|5</math>, the fraction is reducible. There are 249 cases where <math>d|2</math>, 99 where <math>d|5</math>, and 49 where <math>d|(2*5=10)</math>, so by PIE, the number of fails is 299, so our answer is <math>\boxed{200}</math>.
+Say <math>r=\frac{d}{1000-d}</math>; then <math>1<=d<=499</math>. If this fraction is reducible, then the modulus of some number for <math>d</math> is the same as the modulus for <math>1000-d</math>. Since <math>1000=2^3*5^3</math>, that modulus can only be <math>2</math> or <math>5</math>. This implies that if <math>d|2</math> or <math>d|5</math>, the fraction is reducible. There are 249 cases where <math>d|2</math>, 99 where <math>d|5</math>, and 49 where <math>d|(2*5=10)</math>, so by PIE, the number of fails is 299, so our answer is <math>\boxed{200}</math>.
 == See also ==
 {{AIME box|year=2014|n=I|num-b=2|num-a=4}}
 {{MAA Notice}}

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