Fraction reduction, euclid's algorithm

[rayman123]

Homework Statement

reduce the fraction \frac{943578}{1978935} to its lowest terms using Euclid's algorithm

The Attempt at a Solution

I start with finding the gcd of these two numbers using E.algorithm

1978935=943578*2+91779
942578=91779*10+25788
91779=25788*3+14415
25788=14415*1+11373
14415=11373*1+3042
11373=3042*3+2247
3042=2247*1+795
2247=795*2+657
795=657*1+138
657=138*4+105
138=105*1+33
105=33*3+6
33=6*5+3
6=2*3+0
so gcd(1978935,942578)=3
but here I am somehow unable to use it to reduce the fraction. Please help

 

[HallsofIvy]

Divide both numerator and denominator by that "greatest common divisor".

 

[rayman123]

I am not sure if this is the way we are supposed to do it...Euclid's algorithm is to help to make the calculations easier, without calculator no one can do such divisions...
\frac{942578}{1978935}=\frac{314526}{659645}
but how can we be sure that this is not further reducible?