Exercise 929Number theory › Divisibility and gcdOlympiadAdded 3 Nov 2026

The radical of an integer

#radical#squarefree#prime factorization

Statement

Define the radical $\operatorname{rad}(n)=\prod_{p\mid n}p$ (product of distinct prime divisors). Show that $n/\operatorname{rad}(n)$ is a perfect square iff $n$ is of the form $m^2\cdot s$ where $s$ is squarefree and $\operatorname{rad}(s)=\operatorname{rad}(n)$ .

Hint

Write $n=\prod p^{\alpha_p}$ and compute $n/\operatorname{rad}(n)=\prod p^{\alpha_p-1}$ .

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Solution

Computation. $n/\operatorname{rad}(n)=\prod p^{\alpha_p-1}$ .

Perfect square condition. $\alpha_p-1$ even for all $p$ , i.e. $\alpha_p$ odd for all $p\mid n$ . So $n=\prod p^{2\beta_p+1}=(\prod p^{\beta_p})^2\cdot\prod p$ , i.e. $n=m^2\cdot\operatorname{rad}(n)$ where $m=\prod p^{\beta_p}$ .

Conclusion. $n/\operatorname{rad}(n)$ is a perfect square iff all prime exponents in $n$ are odd, iff $n=m^2\operatorname{rad}(n)$ .

The radical of an integer

Statement

Solution

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