If ab, ac and bc are perfect cube, prove a,b,c are also perfect cube.?

Question

If ab, ac and bc are perfect cube, prove a,b,c are also perfect cube.?

Answer 1

Proof by contradiction:

Suppose ab, ac, bc are all cubes and not all of a, b, c are.

Consider the prime factors of a, b, c.

Since we suppose a is not a cube, then at least one of its prime factors

does not occur a multiple of 3 times.

That prime factor must occur either 3n+1 or 3n+2 times, for some n.

But ab is a cube, so that same factor much occur 3m+2 or 3m+1 times in b,

since all factors in ab occur a multiple of three times.

Let's say the factor occurs 3n+1 times in a and 3m+2 times in b.

(if not, just switch a and b)

That same factor must occur either 3j, 3j+1, or 3j+2 times in c.

In any of those cases, it prevents ac or bc or both from being cubes,

which is a contradiction.

Thus all prime factors must occur 3k times in all of a, b, and c,

and they are all perfect cubes.

.

Answer 2

That's only three sides.  Which would make a triangle, not a cube.  If you're gonna ask us to do your homework, at least get the problem correct.