Question

Найдите сумму а+b , если НОД (а,b)=3 для натуральных чисел, удовлетворяющих равенству a/2=b/3 Заранее спасибо решение подробно пожалуйста)

Answer 1

Problem Analysis

We are given that the greatest common divisor (GCD) of two natural numbers, a and b, is 3. We are also given that a/2 = b/3. We need to find the sum of a and b.

Solution

To solve this problem, we can use the relationship between the GCD and the least common multiple (LCM) of two numbers. The product of the GCD and the LCM of two numbers is equal to the product of the two numbers.

Let's assume that the LCM of a and b is lcm(a, b). According to the given information, we have:

a/2 = b/3

This implies that a = 2b/3.

Now, we can express the LCM of a and b in terms of b:

lcm(a, b) = (a * b) / gcd(a, b) = ((2b/3) * b) / 3 = (2b^2) / 9

Since the GCD of a and b is 3, we have:

lcm(a, b) * gcd(a, b) = a * b (2b^2 / 9) * 3 = a * b 2b^2 = 27ab

Simplifying the equation, we get:

2b = 27a

Substituting this value of b in terms of a into the equation a = 2b/3, we can solve for a:

a = 2b/3 a = 2(27a)/3 a = 18a 1 = 18

This equation is not possible, which means there are no natural numbers that satisfy the given conditions. Therefore, there is no solution to this problem.

Conclusion

The sum of a and b cannot be determined because there are no natural numbers that satisfy the given conditions.