Question

[tex]Aratati ca urmatoarele fractii sunt ireductibile :
a) \frac{2n+3}{3n+4}
b) \frac{2n+7}{n+3}
c) \frac{2n+3}{5n+7}
d) \frac{3n+7}{5n+12}

Answer 1

a) 2n+3/3n+4 
Presupunem ca au un divizor comun "d": 
d /2n+3⇒d /6n+9
d/ 3n+4⇒d/6n+8
 Scadem:d/1⇒fractie ireductibila
Singurul numar care se divide cu 1 este chiar 1.
Am folosit proprietatile :Daca d/a si d/b ,atunci d/a-b.
                                      Daca d/a⇒ d/a*k ,k∈N
Celelalte se fac analog:
b)2n+7/n+3
d/2n+7⇒d/2n+7
d/n+3  ⇒d/2n+6
Scadem: d/1⇒fractie ireductibila.

c)2n+3/5n+7
d/2n+3⇒d/10n+15
d/5n+7⇒d/10n+14
Scadem: d/1⇒fractie ireductibila

d)3n+7/5n+12
d/5n+12⇒d/15n+36
d/3n+7 ⇒d/15n+35
Scadem:d/1 ⇒fractie ireductibila