G3.4 Spilt the numbers $1,2, \ldots, 10$ into two groups and let $P_{1}$ be the product of the first group and $P_{2}$ the product of the second group. If $P_{1}$ is a multiple of $P_{2}$, find the minimum value of $\frac{P_{1}}{P_{2}}$.
$P_{1}=k P_{2}$, where $k$ is a positive integer.
$\therefore$ All prime factors of $P_{2}$ can divide $P_{1}$.
$\frac{10}{5}=2,10$ must be a factor of the numerator and 5 must a factor of the denominator
7 is a prime which must be a factor of the numerator.
Among the even numbers $2,4,6,8,10$, there are 8 factors of 2 .
4 factors of 2 should be put in the numerator and 4 factors should be put in the denominator. Among the number $3,6,9$, there are 4 factors of 3 .
2 factors of 3 should be put in the numerator and 2 factors should be put in the denominator.
Minimum value of $\frac{P_{1}}{P_{2}}=\frac{8 \times 7 \times 9 \times 10}{1 \times 2 \times 3 \times 4 \times 5 \times 6}=7$