i need a hint please 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.

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- Feb 15th 2009, 10:36 AM #1

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- Feb 15th 2009, 11:50 AM #2

- Feb 15th 2009, 12:00 PM #3
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 $\displaystyle n=m^2 k$

If k is square free, then you are done : b=mē and a=k

If k is not square free, then there exists integers p>1 and q>1 such that $\displaystyle k=p^2 q$

So $\displaystyle n=(mp)^2 q$

If q is square free...

If q is not square free...

And so on...

- Feb 15th 2009, 01:55 PM #4