3. Find the smallest positive integer $m$ such that $5 m$ is a fifth power of a positive integer, $6 m$ is a sixth power of a positive integer, and $7 m$ is a seventh power of a positive integer.
(2013, Irish Mathematical Olympiad)
Let
$$
m=2^{a} \times 3^{b} \times 5^{c} \times 7^{d} M(a, b, c, d \in \mathbf{N}),
$$
where $M$ is a positive integer that does not contain the prime factors $2, 3, 5, 7$.
By the problem's condition, to make $m$ as small as possible, we take $M=1$.
Since $5m$ is a fifth power of a positive integer, all the exponents of the prime factors of $5m$ must be divisible by 5.
Thus, $a=5a_{1}, b=5b_{1}, c+1=5c_{1}, d=5d_{1}$.
The cases for $6m$ and $7m$ are similar.
Therefore, the smallest positive integer
$$
m=2^{35} \times 3^{35} \times 5^{84} \times 7^{90} \text{. }
$$