multiply all 3 to get (abc)^2 = 576 => abc = 24

ab = 4 = 24/c =>c = 6

then u get b = 9/6 = 3/2 and a = 16/6 = 8/3

So you rearrange one of the equations with c so c is on one side.

c = 9/b

and stick that in for c.

(9/b) * a = 4

Might as well turn that “a” into an expression with b, too. So a=4/b

Then more replacing,

(9/b) * (4/b) = 4

36/ (b^2) = 4

Take the inverse (b^2)/36 = 1/4

b^2 = 9

b= 3

Now just plug in 3 instead of b for the other equations to find a and c.

a=16/c

c=16/a

b=9/c

c=9/b

b=4/a

a=4/b

ab=4

a * (4/a) =4

4=4

1=1

bc=9

b* (9/b) =9

9=9

1=1

ca=16

c* (16/c)=16

16=16

1=1

(abc)^2 =4*9*16

abc= 2*3*4

this implies

a=8/3

b=3/2

c=6