1. ## need help

Show that every positive integer n can be written uniquely in the form n=ab, where a is a square-free and b is a square. Show that b is then the largest square dividing n.

2. Originally Posted by mancillaj3
Show that every positive integer n can be written uniquely in the form n=ab, where a is a square-free and b is a square. Show that b is then the largest square dividing n.
By 'square free', do you mean not divisible by any integer of the $a^2$ with $a>1$?

3. Originally Posted by TheMasterMind
By 'square free', do you mean not divisible by any integer of the $a^2$ with $a>1$?
It's the exact definition of square free

If n is square-free, then a=n and b=1
If n is a square, then a=1 and b=n

If n is not square free, nor a square, it means that there exists integers m>1 and k>1 such that $n=m^2 k$
If k is square free, then you are done : b=m² and a=k
If k is not square free, then there exists integers p>1 and q>1 such that $k=p^2 q$
So $n=(mp)^2 q$

If q is square free...
If q is not square free...

And so on...

4. Originally Posted by Moo
It's the exact definition of square free

If n is square-free, then a=n and b=1
If n is a square, then a=1 and b=n

If n is not square free, nor a square, it means that there exists integers m>1 and k>1 such that $n=m^2 k$
If k is square free, then you are done : b=m² and a=k
If k is not square free, then there exists integers p>1 and q>1 such that $k=p^2 q$
So $n=(mp)^2 q$

If q is square free...
If q is not square free...

And so on...
Very nice, I was thinking of something along those lines but wanted to check the definition first, make sure my memory is not deceiving me!