Estonian Math Competitions

Let $n=2^{\alpha }\cdot 3^{\beta }\cdot 5^{\gamma }\cdot 7^{\delta }\cdot s$, where $s$ is not divisible by $2$, $3$, $5$ or $7$; then $5n=2^{\alpha }\cdot 3^{\beta }\cdot 5^{\gamma +1}\cdot 7^{\delta }\cdot s$, $6n=2^{\alpha +1}\cdot 3^{\beta +1}\cdot 5^{\gamma }\cdot 7^{\delta }\cdot s$ and $7n=2^{\alpha }\cdot 3^{\beta }\cdot 5^{\gamma }\cdot 7^{\delta +1}\cdot s$. Consequently: * For $\sqrt[5]{5n}$ to be an integer, $\alpha$, $\beta$, $\gamma +1$ and $\delta$ must be divisible by $5$; * For $\sqrt[6]{6n}$ to be an integer, $\alpha +1$, $\beta +1$, $\gamma$ and $\delta$ must be divisible by $6$; * For $\sqrt[7]{7n}$ to be an integer, $\alpha$, $\beta$, $\gamma$ and $\delta +1$ must be divisible by $7$. Hence $\alpha$ and $\beta$ must be divisible by $35$, $\gamma$ must be divisible by $42$ and $\delta$ must be divisible by $30$. The least suitable value for $\alpha$ and $\beta$ is $35$ since $35$ is the least positive multiple of $35$ and the next integer is divisible by $6$. Studying the multiples of $42$ and $30$ similarly shows that the least suitable value for $\gamma$ is $84$ and the least suitable value for $\delta$ is $90$. For the least suitable value for $n$, take $s=1$. Hence the desired number is $2^{35}\cdot 3^{35}\cdot 5^{84}\cdot 7^{90}$.