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I need to show that every positive integer $n$ can be written uniquely in the form $n = ab$, where $a$ is square-free and $b$ is a square. Then I need to show that $b$ is then the largest square dividing $n$.

The problem I have here is that I can't even see how this is true. How can $1$ be represented this way? Is $1$ a square number? If not, then I cannot see how this is possible. If it is, then I cannot see how $1$, $2$, or $3$ can be represented this way.

Please, any help you can offer giving me a push in the right direction here is greatly appreciated.

  • 0
    1 is a square, 1 = 1*1, 2 = 2*1, 3 = 3*12011-02-09
  • 6
    $1$ is both square-free and a square.2011-02-09

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