What is the least positive integer x for which 12x is
the cube of an integer?

Question

What is the least positive integer x for which 12x is
the cube of an integer?

anonymous · Answer

@phi

kinggeorge · Answer

First, you'll notice that $12=2^2\cdot3$. So if $12x=n^3$ for some integer $n$, then $3$ must divide $n^3$, so $3$ must divide $n$. Make sense so far?

phi · Answer

write 12x as 
2*2*3*x
now you need 3 copies of everything... 3 2's   and three 3's
what should x be ?

anonymous · Answer

so 2*2*2*3*3*3*x?

phi · Answer

you want x so that you end up with 2*2*2*3*3*3

anonymous · Answer

ok totaly confused. 2*2*2*3*3*3= 216?

phi · Answer

the idea is if you have
2*2*3 * more_stuff
to make that a "cube" you want 3 of each number.
you have 2*2 , and if you multiply by 2 you would get
2*2*2* 3
but you need two more 3's, so you should multiply by 3*3 also
all together  you multiply by some number to change
2*2*3  to 2*2*2*3*3*3 .  what number is that ?

phi · Answer

I meant , what number do we multiply 2*2*3  to get 2*2*2*3*3*3 ?

don't you think we should multiply by 2*3*3 ?

anonymous · Answer

no?

phi · Answer

Let's start over

we have
2*2*3*x
we want 2*2*2* 3*3*3
what is x?

anonymous · Answer

2*3*3?

phi · Answer

yes,  now change that into a number = 18

x is 18

anonymous · Answer

so the answer would be 18?

phi · Answer

yes

anonymous · Answer

THANK YOU!! can u help with one more?