【Analysis】First, decompose the large number into the product of multiple smaller numbers. Within the range of 1 to 30, there are many identical smaller integers. To find the minimum value of the quotient of two groups of numbers, it means the products of the two groups of numbers are very close. Distribute the identical smaller numbers evenly between the two groups, and place the remaining single smaller numbers into one of the groups. Calculate the minimum value of $\frac{A}{B}$.

【Solution】Solution: The prime numbers between 1 and 30 are $2,3,5,7,11,13,17,19,23,29$, and the composite numbers are: $4=2 \times 2,6=2 \times 3,8=2 \times 2 \times 2,9=3 \times 3,10=2 \times 5,12=2 \times 2 \times 3,14=2 \times 7$, $15=3 \times 5,16=2 \times 2 \times 2 \times 2,18=2 \times 3 \times 3$, $20=2 \times 2 \times 5,21=3 \times 7,22=2 \times 11,24=2 \times 2 \times 2 \times 3,25=5 \times 5,26=2 \times 13,27=3$ $\times 3 \times 3,28=2 \times 2 \times 7,30=2 \times 3 \times 5$,

Organize all the factors: there are 26 twos, 14 threes, 7 fives, 4 sevens, 2 elevens, 2 thirteens, 1 seventeen, 1 nineteen, 1 twenty-three, and 1 twenty-nine, i.e., $30!=2^{26} \times 3^{14} \times 5^{7} \times 7^{4} \times 11^{2} \times 13^{2} \times 17 \times 19 \times 23 \times$ 29,

To make $\frac{\mathrm{A}}{\mathrm{B}}$ as small as possible, the prime factors should be distributed as evenly as possible, so $\frac{\mathrm{A}}{\mathrm{B}}=5 \times 17 \times 19 \times 23 \times 29=1077205$.
Answer: The minimum value of $\frac{\mathrm{A}}{\mathrm{B}}$ is 1077205.
13. (15 points) Divide the natural numbers from 1 to 30 into two groups, such that the product of all numbers in the first group \( A \) is divisible by the product of all numbers in the second group \( B \). What is the minimum value of \( \frac{A}{B} \)?