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Difference between revisions of "2022 AIME I Problems/Problem 13"

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Let <math>S</math> be the set of all rational numbers that can be expressed as a repeating decimal in the form <math>0.\overline{abcd},</math> where at least one of the digits <math>a,</math> <math>b,</math> <math>c,</math> or <math>d</math> is nonzero. Let <math>N</math> be the number of distinct numerators obtained when numbers in <math>S</math> are written as fractions in lowest terms. For example, both <math>4</math> and <math>410</math> are counted among the distinct numerators for numbers in <math>S</math> because <math>0.\overline{3636} = \frac{4}{11}</math> and <math>0.\overline{1230} = \frac{410}{3333}.</math> Find the remainder when <math>N</math> is divided by <math>1000.</math>
 
Let <math>S</math> be the set of all rational numbers that can be expressed as a repeating decimal in the form <math>0.\overline{abcd},</math> where at least one of the digits <math>a,</math> <math>b,</math> <math>c,</math> or <math>d</math> is nonzero. Let <math>N</math> be the number of distinct numerators obtained when numbers in <math>S</math> are written as fractions in lowest terms. For example, both <math>4</math> and <math>410</math> are counted among the distinct numerators for numbers in <math>S</math> because <math>0.\overline{3636} = \frac{4}{11}</math> and <math>0.\overline{1230} = \frac{410}{3333}.</math> Find the remainder when <math>N</math> is divided by <math>1000.</math>
  
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==Solution==
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==Solution 1==
  
 
<math>0.\overline{abcd}=\frac{abcd}{9999} = \frac{x}{y}</math>, <math>9999=9\times 11\times 101</math>.
 
<math>0.\overline{abcd}=\frac{abcd}{9999} = \frac{x}{y}</math>, <math>9999=9\times 11\times 101</math>.
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'''mathboy282'''
 
'''mathboy282'''
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==Solution 2 (similar''(?)'' to solution 1)==
  
 
==Video Solution==
 
==Video Solution==

Revision as of 20:45, 2 January 2024

Problem

Let $S$ be the set of all rational numbers that can be expressed as a repeating decimal in the form $0.\overline{abcd},$ where at least one of the digits $a,$ $b,$ $c,$ or $d$ is nonzero. Let $N$ be the number of distinct numerators obtained when numbers in $S$ are written as fractions in lowest terms. For example, both $4$ and $410$ are counted among the distinct numerators for numbers in $S$ because $0.\overline{3636} = \frac{4}{11}$ and $0.\overline{1230} = \frac{410}{3333}.$ Find the remainder when $N$ is divided by $1000.$

Solution 1

$0.\overline{abcd}=\frac{abcd}{9999} = \frac{x}{y}$, $9999=9\times 11\times 101$.

Then we need to find the number of positive integers $x$ that (with one of more $y$ such that $y|9999$) can meet the requirement $1 \leq {x}\cdot\frac{9999}{y} \leq 9999$.

Make cases by factors of $x$. (A venn diagram of cases would be nice here.)


Case $A$:

$3 \nmid x$ and $11 \nmid x$ and $101 \nmid x$, aka $\gcd (9999, x)=1$.

Euler's totient function counts these: \[\varphi \left(3^2 \cdot 11 \cdot 101 \right) = ((3-1)\cdot 3)(11-1)(101-1)= \bf{6000}\] values (but it's enough to note that it's a multiple of 1000 and thus does not contribute to the final answer)

Note: You don't need to know this formula. The remaining cases essentially re-derive the same computation for other factors of $9999$. This case isn't actually different.

The remaining cases have $3$ (or $9$), $11$, and/or $101$ as factors of $abcd$, which cancel out part of $9999$. Note: Take care about when to use $3$ vs $9$.


Case $B$: $3|x$, but $11 \nmid x$ and $101 \nmid x$.

Then $abcd=9x$ to leave 3 uncancelled, and $x=3p$, so $x \leq \frac{9999}{9} = 1111$, giving:

$x \in 3 \cdot \{1, \dots \left\lfloor \frac{1111}{3}\right\rfloor\}$,

$x \notin (3\cdot 11) \cdot \{1 \dots \left\lfloor \frac{1111}{3\cdot 11}\right\rfloor\}$,

$x \notin (3 \cdot 101) \cdot \{1 \dots \left\lfloor \frac{1111}{3 \cdot 101}\right\rfloor\}$,

for a subtotal of $\left\lfloor \frac{1111}{3}\right\rfloor - (\left\lfloor\frac{1111}{3 \cdot 11}\right\rfloor + \left\lfloor\frac{1111}{3 \cdot 101}\right\rfloor ) = 370 - (33+3) = \bf{334}$ values.


Case $C$: $11|x$, but $3 \nmid x$ and $101 \nmid x$.

Much like previous case, $abcd$ is $11x$, so $x \leq \frac{9999}{11} = 909$,

giving $\left\lfloor \frac{909}{11}\right\rfloor - \left(\left\lfloor\frac{909}{11 \cdot 3}\right\rfloor + \left\lfloor\frac{909}{11 \cdot 101}\right\rfloor \right) = 82 - (27 + 0) = \bf{55}$ values.


Case $D$: $3|x$ and $11|x$ (so $33|x$), but $101 \nmid x$.

Here, $abcd$ is $99x$, so $x \leq \frac{9999}{99} = 101$,

giving $\left\lfloor \frac{101}{33}\right\rfloor - \left\lfloor \frac{101}{33 \cdot 101}\right\rfloor = 3-0 = \bf{3}$ values.


Case $E$: $101|x$.

Here, $abcd$ is $101x$, so $x \leq \frac{9999}{101} = 99$,

giving $\left\lfloor \frac{99}{101}\right\rfloor = \bf{0}$ values, so we don't need to account for multiples of $3$ and $11$.

To sum up, the answer is \[6000+334+55+3+0\equiv\boxed{392} \pmod{1000}.\]

Clarification

In this context, when the solution says, "Then $abcd=9x$ to leave 3 uncancelled, and $x=3p$," it is a bit vague. The best way to clarify this is by this exact example - what is really meant is we need to divide by 9 first to achieve 1111, which has no multiple of 3; thus, given that the fraction x/y is the simplest form, x can be a multiple of 3.

Similar explanations can be said when the solution divides 9999 by 11, 101, and uses that divided result in the PIE calculation rather than 9999.

mathboy282

Solution 2 (similar(?) to solution 1)

Video Solution

https://youtu.be/0FZyjuIOHnA

https://MathProblemSolvingSkills.com

See Also

2022 AIME I (Problems • Answer Key • Resources)
Preceded by
Problem 12 		Followed by
Problem 14
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All AIME Problems and Solutions

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