定理4:)若fgh,且(f,g)=1,则fh i)若fg,f2|g,且(f1,2)=1,则f/2|g i)若(f,g)=1,(f,h)=1,则(f,gh)=1. proof:i)由uf+vg=1,两边同乘h fh+vgh=h→>fh i)g=ih1,/21h,且(f1,)=1,应2h1 内1=/2h2,g=/2h2,/2 i)由l+wg=1,sf+th=1,两边相乘得 uf(sf+th)+sfvg+vtgh=1 (f, gh)10 ) ( , ) 1, ( , ) 1, ( , ) 1. ) , , ( , ) 1, . 4 ) , ( , ) 1, . 1 2 1 2 1 2 = = = = = iii f g f h f gh ii f g f g f f f f g i f gh f g f h 若 则 若 且 则 定理 ： 若 且 则 . : ) 1, u f h v g h h f h proof i u f v g h + = → 由 + = 两边同乘 , , . ) , , ( , ) 1, 1 2 2 1 2 2 1 2 1 1 2 1 1 1 2 2 1 h f h g f f h f f g ii g f h f f h f f f h  =  = = 且 =  ( , ) 1. ( ) 1 ) 1, 1,  = + + + = + = + = f gh uf sf t h sfvg vtgh iii 由uf v g sf t h 两边相乘得