The natural numbers from 1 to 30 are divided into two groups, so that the product a of all numbers in the first group can be divided by the product B of all numbers in the second group. What is the minimum value of AB?

The natural numbers from 1 to 30 are divided into two groups, so that the product a of all numbers in the first group can be divided by the product B of all numbers in the second group. What is the minimum value of AB?

Among the natural numbers from 1 to 30, prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, which are 4 = 2 × 2, 6 = 2 × 3, 8 = 2 × 2 × 2, 9 = 3 × 3, 10 = 2 × 5, 12 = 2 × 2 × 3, 14 = 2 × 7, 15 = 3 × 5, 16 = 2 × 2 × 2, 18 = 2 × 3 × 3, 20 = 2 × 2 × 5, 21 = 3 × 7, 22 = 2 × 11, 24 = 2

A two digit natural number can be divided by its number product to find the natural number

Let ten digits be a and one digit be B, then 10A + B can be divided by ab
10A + B can be divisible by a, indicating that B can be divisible by a, assuming that B = Ka, K

It is proved by mathematical induction that the product of two continuous positive integers can be divisible by 2

1) When continuous positive integers are 1 and 2, 1 × 2 = 2 can be divisible by 2
2) Suppose that the product K (K + 1) of K and K + 1 can be divisible by 2,
Then the product of K + 1 and K + 2 (K + 1) (K + 2) = K (K + 1) + 2 (K + 1)
∵ K (K + 1) and 2 (K + 1) are divisible by 2
That is, (K + 1) (K + 2) can also be divisible by 2
It is known from 1) and 2) that the product of two consecutive positive integers can be divisible by 2