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Difference between revisions of "1959 IMO Problems/Problem 1"
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As in the first solution, it follows that <math>\frac{21n+4}{14n+3}</math> is irreducible.  Q.E.D.
 
As in the first solution, it follows that <math>\frac{21n+4}{14n+3}</math> is irreducible.  Q.E.D.
  
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=== Third Solution ===
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[[Proof by contradiction]]:
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Let's assume that <math>\dfrac{14n+3}{21n+4}</math> is a [[reducible fraction]] where <math>p</math> is a [[divisor]] of both the [[numerator]] and the [[denominator]]:
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<center>
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<math>14n+3\equiv 0\pmod{p} \implies 42n+9\equiv 0\pmod{p}</math>
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<math>21n+4\equiv 0\pmod{p} \implies 42n+8\equiv 0\pmod{p}</math>
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</center>
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Subtracting the second [[equation]] from the first [[equation]] we get <math>1\equiv 0\pmod{p}</math> which is cleary absurd.
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Hence <math>\frac{21n+4}{14n+3}</math> is irreducible.  Q.E.D.
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<!--Solution by tonypr-->
  
 
{{IMO box|year=1959|before=First question|num-a=2}}
 
{{IMO box|year=1959|before=First question|num-a=2}}

Revision as of 21:44, 17 May 2009

Problem

Prove that the fraction $\frac{21n+4}{14n+3}$ is irreducible for every natural number $n$.


Solutions

First Solution

We observe that

$3(14n+3) = 2(21n+4) + 1.$


Since a multiple of $14n+3$ differs from a multiple of $21n+4$ by 1, we cannot have any postive integer greater than 1 simultaneously divide $14n+3$ and $21n+4$. Hence the greatest common divisor of the fraction's numerator and denominator is 1, so the fraction is irreducible. Q.E.D.

Second Solution

Denoting the greatest common divisor of $a, b$ as $(a,b)$, we use the Euclidean algorithm as follows:

$( 21n+4, 14n+3 ) = ( 7n+1, 14n+3 ) = ( 7n+1, 1 ) = 1$

As in the first solution, it follows that $\frac{21n+4}{14n+3}$ is irreducible. Q.E.D.

Third Solution

Proof by contradiction:

Let's assume that $\dfrac{14n+3}{21n+4}$ is a reducible fraction where $p$ is a divisor of both the numerator and the denominator:

$14n+3\equiv 0\pmod{p} \implies 42n+9\equiv 0\pmod{p}$

$21n+4\equiv 0\pmod{p} \implies 42n+8\equiv 0\pmod{p}$

Subtracting the second equation from the first equation we get $1\equiv 0\pmod{p}$ which is cleary absurd.

Hence $\frac{21n+4}{14n+3}$ is irreducible. Q.E.D.

1959 IMO (Problems) • Resources
Preceded by
First question 		1 • 2 • 3 • 4 • 5 • 6 Followed by
Problem 2
All IMO Problems and Solutions
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