Question

З 22 карток на яких записані числа від 1 до 22 склали 11 дроб. Яка найбільша кілк. отриманих дробів може = цілому числу?

Answer 1

22 21 19 15 16 14 18 12 20        

11 3    1    5   8   7   9    6 10

Числа 13 і 17 і19 прості. Отже для них підходить тільки цифра 1 тому з 11 дробів 9 можуть дорівнювати цілому числу

Answer 2

Finding the Maximum Number of Fractions That Can Sum to an Integer

To find the maximum number of fractions that can sum to an integer from 22 cards with numbers from 1 to 22, we can consider the concept of Egyptian fractions. Egyptian fractions are unit fractions (fractions with 1 in the numerator) that sum to a given number. Let's explore this further.

Egyptian Fractions

Egyptian fractions are representations of positive rational numbers as sums of distinct unit fractions. A unit fraction is a fraction with 1 in the numerator. The ancient Egyptians used this method for representing fractions. The problem of representing a given fraction as a sum of distinct unit fractions is known as the Egyptian Fraction Problem.

Maximum Number of Fractions

To find the maximum number of fractions that can sum to an integer, we can use the concept of Egyptian fractions. The maximum number of fractions that can sum to an integer is equal to the numerator of the given fraction. In this case, the given fraction is the sum of the unit fractions.

Calculation

Given that there are 22 cards with numbers from 1 to 22, we want to find the maximum number of fractions that can sum to an integer. To do this, we need to find the sum of distinct unit fractions that equals an integer.

The maximum number of fractions that can sum to an integer from 22 cards with numbers from 1 to 22 is 21. This is because the sum of the unit fractions can be represented as 1/2 + 1/3 + 1/4 + ... + 1/22, which equals 21.

Therefore, the maximum number of fractions that can sum to an integer from the given set of cards is 21.