Difference between revisions of "1959 IMO Problems/Problem 1"
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== Solutions == | == Solutions == | ||
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Denoting the greatest common divisor of <math>a, b </math> as <math>(a,b) </math>, we use the [[Euclidean algorithm]] as follows: | Denoting the greatest common divisor of <math>a, b </math> as <math>(a,b) </math>, we use the [[Euclidean algorithm]] as follows: | ||
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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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[[Proof by contradiction]]: | [[Proof by contradiction]]: | ||
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We notice that: | We notice that: | ||
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By Bezout's Theorem, <math>3 \cdot (14n+3) - 2 \cdot (21n + 4) = 1</math>, so the gcd of the numerator and denominator is 1 and the fraction is irreducible. | By Bezout's Theorem, <math>3 \cdot (14n+3) - 2 \cdot (21n + 4) = 1</math>, so the gcd of the numerator and denominator is 1 and the fraction is irreducible. | ||
Revision as of 13:16, 25 May 2018
Contents
[hide]Problem
Prove that the fraction is irreducible for every natural number .
Solutions
Solution
Denoting the greatest common divisor of as , we use the Euclidean algorithm as follows:
As in the first solution, it follows that is irreducible. Q.E.D.
Solution 2
Assume that is a reducible fraction where is a divisor of both the numerator and the denominator:
Subtracting the second equation from the first equation we get which is clearly absurd.
Hence is irreducible. Q.E.D.
Solution 3
We notice that:
So it follows that and must be coprime for every natural number for the fraction to be irreducible. Now the problem simplifies to proving irreducible. We re-write this fraction as:
Since the denominator differs from a multiple of the numerator by 1, the numerator and the denominator must be relatively prime natural numbers. Hence it follows that is irreducible.
Q.E.D
Solution 4
By Bezout's Theorem, , so the gcd of the numerator and denominator is 1 and the fraction is irreducible.
Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.
See Also
1959 IMO (Problems) • Resources | ||
Preceded by First question |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 2 |
All IMO Problems and Solutions |