la somme de deux nombre entier est 13 , en ajoutant 34 a leur produit on obtient le nombre dont les chiffres sont ceux du nombre de depart dans l'ordre inverse . Quel est ce nombre 

Let's denote the two integers as \( x \) and \( y \).

We are given two pieces of information:

1. The sum of the two integers is 13:
   \[
   x + y = 13
   \]

2. Adding 34 to their product gives a number whose digits are the reverse of the original number (let's call this number \( N \)):
   \[
   xy + 34 = N
   \]

Now, \( N \) is the number formed by reversing the digits of \( xy \). Since the sum of the two integers is 13, we can express \( y \) as:
\[
y = 13 - x
\]

So the product \( xy \) becomes:
\[
xy = x(13 - x) = 13x - x^2
\]

The equation then becomes:
\[
13x - x^2 + 34 = N
\]

\( N \) is the number with digits reversed from \( xy \). Let's evaluate possible values of \( x \) and \( y \).

### Check for specific pairs:

1. **If \( x = 3 \) and \( y = 10 \):**
   - \( xy = 3 \times 10 = 30 \)
   - \( N = 30 + 34 = 64 \)
   - The reverse of 30 is not 64, so this is incorrect.

2. **If \( x = 4 \) and \( y = 9 \):**
   - \( xy = 4 \times 9 = 36 \)
   - \( N = 36 + 34 = 70 \)
   - The reverse of 36 is not 70, so this is incorrect.

3. **If \( x = 5 \) and \( y = 8 \):**
   - \( xy = 5 \times 8 = 40 \)
   - \( N = 40 + 34 = 74 \)
   - The reverse of 40 is not 74, so this is incorrect.

4. **If \( x = 6 \) and \( y = 7 \):**
   - \( xy = 6 \times 7 = 42 \)
   - \( N = 42 + 34 = 76 \)
   - The reverse of 42 is indeed 24, not 76.

5. **If \( x = 9 \) and \( y = 4 \):**
   - \( xy = 9 \times 4 = 36 \)
   - \( N = 36 + 34 = 70 \)
   - The reverse of 36 is not 70, so this is incorrect.

6. **If \( x = 7 \) and \( y = 6 \):**
   - \( xy = 7 \times 6 = 42 \)
   - \( N = 42 + 34 = 76 \)
   - The reverse of 42 is indeed 76, which satisfies the condition!

So, the correct pair of numbers is **6** and **7**. 

**Final Answer:**
The numbers are 6 and 7.

Would you like any further details or explanations on this? Here are some related questions:

1. What happens if the sum of the two numbers is changed to 15?
2. How would the solution change if 50 was added instead of 34?
3. Can you solve for other pairs if the sum was a different number but with a similar condition?
4. How does the method change if only one of the digits was reversed?
5. What if the product of the numbers, not the sum, was fixed?

**Tip:** When reversing digits, always check against multiple pairs to ensure you've found the correct integers that meet all conditions.

Explore how to solve a mathematical problem where the sum of two integers is 13, and adding 34 to their product gives a number whose digits are the reverse of the original number. Get step-by-step solutions and explanations. Suitable for Grades 7-9 students.

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