Please give me help to find out the answer for following question
With the usual notation show that
i. For a > 0, and b > 0,[a,b](a,b) = ab
ii. [a,b, c](ab,bc, ca) = abc if a,b, c are positive integers.
(i) let (a,b)=d,then there exist positive integers m and n such that a=md,b=nd,and (m,n)=1.
Thus [a,b]=mnd; ab=mdnd; => [a,b](a,b)=ab. Q.E.D
(ii)By (i)(proved), we have:
$\displaystyle [a,b,c]=[\frac{ab}{(a,b)},c]=\frac{\frac{abc}{(a,b)}}{(\frac{ab}{(a,b)},c)}=\f rac{abc}{(a,b)(\frac{ab}{(a,b)},c)}=\frac{abc}{(ab ,bc,ac)}$
Thus [a,b,c](ab,bc,ac)=abc; Q.E.D