Question

Сумма двух натуральных чисел равна 1001. Какое наибольшее значение может принимать их наибольший общий делитель?

Answer 1

Пусть n - наибольший общий делитель.
Тогда выражение a+b=1001 можно записать в виде:
 pn+qn=1001 или:
 n(p+q)=1001.
Очевидно, что n находится среди делителей числа 1001. Разложим его на простые множители: 1001=7*11*13. В силу натуральности чисел a и b, ни одно из них не равно 0.
Поэтому НОД (a, b)=11*13=143, а наши числа 143 и 858

Answer 2

Solving for the Greatest Common Divisor (GCD) of Two Natural Numbers

To find the greatest value of the greatest common divisor (GCD) of two natural numbers whose sum is 1001, we can use the fact that the GCD of two numbers divides their sum. Therefore, the greatest possible value for the GCD occurs when the two numbers are as close as possible to each other.

Let's denote the two natural numbers as a and b, such that a + b = 1001.

Finding the Maximum GCD

The maximum value for the GCD of two natural numbers whose sum is 1001 occurs when the two numbers are as close as possible to each other. This happens when the two numbers are half of the sum, i.e., a = 500 and b = 501.

Now, we can calculate the GCD of 500 and 501 to find the maximum value of the GCD.

Calculation

The GCD of 500 and 501 can be calculated using various methods, such as the Euclidean algorithm or prime factorization. Using the Euclidean algorithm: ``` GCD(500, 501) = GCD(501, 500 % 501) = GCD(501, 500) = GCD(500, 501 - 500) = GCD(500, 1) = 1 ``` Therefore, the maximum value of the GCD of two natural numbers whose sum is 1001 is 1.

This calculation confirms that the GCD of 500 and 501 is indeed 1, which is the maximum possible value for the GCD of two natural numbers whose sum is 1001.