Question

На доске числа 1 2 3 4 5 6 7 8 9 10 какое наименьшее количество чисел надо убрать с доски чтобы оставшиеся числа можно было поделить на 2 группы так чтобы произведение в всех чисел группах были равны

Answer 1

Можно: 1) 1+9 2) 6+4

Answer 2

Problem Analysis

To solve this problem, we need to find the smallest number of integers that must be removed from the set {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} so that the remaining numbers can be divided into two groups with equal products.

Solution

To find the solution, we can start by examining the factors of each number in the set and then determine the minimum number of integers that need to be removed.

1. Factor Analysis: We can start by analyzing the factors of each number in the set to identify any common factors that could be used to balance the product of the two groups.

2. Product Balancing: After identifying the common factors, we can then determine the minimum number of integers that need to be removed to ensure that the product of the two groups is equal.

Factor Analysis

Let's analyze the factors of each number in the set {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} to identify any common factors that could help balance the product of the two groups.

- Number 1: Factors: 1 - Number 2: Factors: 1, 2 - Number 3: Factors: 1, 3 - Number 4: Factors: 1, 2, 4 - Number 5: Factors: 1, 5 - Number 6: Factors: 1, 2, 3, 6 - Number 7: Factors: 1, 7 - Number 8: Factors: 1, 2, 4, 8 - Number 9: Factors: 1, 3, 9 - Number 10: Factors: 1, 2, 5, 10

Product Balancing

After analyzing the factors, we can see that the numbers 2 and 3 are common factors for several numbers in the set. To balance the product of the two groups, we need to remove the number 2 or 3 to ensure that the product of the two groups is equal.

Therefore, the minimum number of integers that need to be removed from the set {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} is 1.

This means that by removing either the number 2 or 3, the remaining numbers can be divided into two groups with equal products.

Conclusion

By removing either the number 2 or 3 from the set {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, the remaining numbers can be divided into two groups with equal products.

I hope this helps! If you have further questions or need additional assistance, feel free to ask.