To solve the problem, let's denote the two-digit number as \( 10a + b \), where \( a \) and \( b \) are the digits of the number. We are given two conditions:
1. The sum of the digits is 6.
2. The product of the number and the number obtained by interchanging the digits is 765.
Let's break down the steps to find the numbers:
### Step 1: Set up the equations
Firstly, from the sum of the digits, we have:
[tex]\[ a + b = 6 \][/tex]
Secondly, from the product of the number and its reverse, we have:
[tex]\[ (10a + b) \times (10b + a) = 765 \][/tex]
### Step 2: Solve the system of equations
We now have a system of two equations with two unknowns:
[tex]\[ a + b = 6 \][/tex]
[tex]\[ (10a + b) \times (10b + a) = 765 \][/tex]
### Step 3: Identify Possible Solutions
Given the mathematical complexity, we'll proceed directly to evaluate possible combinations of digits (0-9) that satisfy both equations.
### Step 4: Check for Valid Solutions
Let's test the possible combinations that satisfy the first equation \(a + b = 6\):
1. If \(a = 1\), then \(b = 5\):
Substituting into the second equation:
[tex]\[
(10 \times 1 + 5) \times (10 \times 5 + 1) = 15 \times 51 = 765
\][/tex]
Which is correct.
2. If \(a = 5\), then \(b = 1\):
Substituting into the second equation:
[tex]\[
(10 \times 5 + 1) \times (10 \times 1 + 5) = 51 \times 15 = 765
\][/tex]
Which is also correct.
### Step 5: Conclude the Solution
Thus, the two numbers that satisfy the given conditions are 15 and 51. Checking both conditions for each number confirms that the solutions are correct.
Therefore, the two-digit numbers that satisfy the problem are:
[tex]\[ \boxed{15 \text{ and } 51} \][/tex]
