Irreducibility of a Fraction (1 Viewer)

Sy123 · Oct 28, 2012

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

Irreducibility of a Fraction

Is a proof by induction efficient?

deswa1 · Oct 28, 2012

Just making sure I've got the terminology right- just irreducible simply mean that there are no common factors in the numerator and denominator? So basically you can't make the fraction simpler? i.e. 2/4=1/2

Sy123 · Oct 28, 2012

deswa1 said:
Just making sure I've got the terminology right- just irreducible simply mean that there are no common factors in the numerator and denominator? So basically you can't make the fraction simpler? i.e. 2/4=1/2

Yes

shongaponga · Oct 28, 2012

Don't think we need induction here.

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 positive 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.

Sy123 · Oct 28, 2012

shongaponga said:
Don't think we need induction here.

We observe that $3(14n + 3) = 2(21n +4) + 1$

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

Alright nice observation there, thanks

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Irreducibility of a Fraction (1 Viewer)

Sy123

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deswa1

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Sy123

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shongaponga

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Sy123

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