Questions: Сколько существует таких троек натуральных чисел (A, B, N), что A+B=46, а B больше A ровно на N процентов?

Transcript text: Сколько существует таких троек натуральных чисел $(A, B, N)$, что $A+B=46$, а $B$ больше $A$ ровно на $N$ процентов?

Solution

Solution Steps

To solve this problem, we need to find the number of triples $(A, B, N)$ such that $A + B = 46$ and $B$ is greater than $A$ by exactly $N$ percent. This can be translated into the equation $B = A + \frac{N}{100} \times A$. We can then iterate through possible values of $A$ and calculate $B$ and $N$ to check if they satisfy the given conditions.

Step 1: Define the Problem

We need to find the number of triples of natural numbers $(A, B, N)$ such that:

$A + B = 46$
$B$ is greater than $A$ by $N$ percent, which can be expressed as $B = A + \frac{N}{100} \cdot A$.

Step 2: Express $B$ in Terms of $A$ and $N$

From the equation $A + B = 46$, we can express $B$ as: \[ B = 46 - A \] Substituting this into the second condition gives: \[ 46 - A = A + \frac{N}{100} \cdot A \] This simplifies to: \[ 46 = 2A + \frac{N}{100} \cdot A \] Factoring out $A$ yields: \[ 46 = A \left(2 + \frac{N}{100}\right) \]

Step 3: Solve for $N$

Rearranging the equation gives: \[ A = \frac{46}{2 + \frac{N}{100}} \] To ensure $A$ is a natural number, $2 + \frac{N}{100}$ must be a divisor of 46. The divisors of 46 are $1, 2, 23, 46$.

Step 4: Calculate Possible Values of $N$

For each divisor $d$ of 46, we can set: \[ 2 + \frac{N}{100} = d \implies \frac{N}{100} = d - 2 \implies N = 100(d - 2) \] We will check each divisor to see if it results in a positive integer $N$.

Step 5: List Valid Divisors and Calculate $N$