a number of two digits divided by the sum of the digits the quotient is 7 and the remainder is 6 . if the digits of the number are interchanged, the resulting number exceeds three times the sum of the digits by 5.

let the unit digit be x

let the tens digit by y
the number is 10y + x
the number reversed is 10x + y

translate the English to Math
(Math is a language)

"divided by the sum of the digits the quotient is 7 and the remainder is 6 "

(10y+x)/(x+y) = 7 + 6/(x+y)
times (x+y)
10y + x = 7(x+y) + 6
10y + x = 7x + 7y + 6
3y - 6x = 6
y - 2x = 2
y = 2x+2

" if the digits of the number are interchanged, the resulting number exceeds three times the sum of the digits by 5"
10x + y = 3(x+y) + 5
10x + y = 3x + 3y + 5

Time for you to take over

Oh boy, solving math problems can be a little tricky, but I'll give it a shot! Let's call the tens digit x and the ones digit y.

According to the problem, we can set up the equation:
10x + y = 7(x + y) + 6

Now, let's solve for x + y:
10x + y = 7x + 7y + 6
3x - 6y = 6

Great! We have the first equation. Now let's tackle the second part.

When the digits are interchanged, the resulting number is 10y + x, and it exceeds three times the sum of the digits (3(x + y)) by 5. So we set up the equation:
10y + x = 3(x + y) + 5

Now, let's simplify:
10y + x = 3x + 3y + 5
2y - 2x = 5

Now, we have a system of equations to solve:
3x - 6y = 6
2y - 2x = 5

I hope you have your clown shoes on because here comes the funny part... I'll be right back with the solution! *inserts clownish shenanigans*

Let's solve this step-by-step:

Step 1: Let's assume the number of two digits as '10a + b', where 'a' represents the tens digit and 'b' represents the ones digit.

Step 2: According to the given information, the number divided by the sum of its digits results in a quotient of 7 and a remainder of 6. Mathematically, this can be written as:
(10a + b) รท (a + b) = 7 + 6/(a + b)

Step 3: Solving the equation, we get:
10a + b = 7(a + b) + 6
10a + b = 7a + 7b + 6
3a - 6b = 6

Step 4: Now, let's look at the second part of the given information, where the digits are interchanged and the resulting number exceeds three times the sum of the digits by 5. Mathematically, this can be written as:
(10b + a) - 3(a + b) = 5

Step 5: Solving the equation, we get:
10b + a - 3a - 3b = 5
7b - 2a = 5

Step 6: Now, let's solve the system of equations formed by Step 3 and Step 5 to find the values of 'a' and 'b'.

Step 7: Multiplying the equation from Step 3 by 7 and the equation from Step 5 by 3, we get:
21a - 42b = 42
21b - 6a = 15

Step 8: Adding the equations from Step 7, we get:
15a = 57
a = 57/15
a = 3

Step 9: Substituting the value of 'a' into the equation from Step 3, we get:
3(3) - 6b = 6
9 - 6b = 6
-6b = 6 - 9
-6b = -3
b = -3/-6
b = 0.5

However, since 'b' represents the ones digit, it cannot be a decimal. Therefore, there is no solution for this problem.

Note: It's possible that there may be an error in the problem statement or the question was written incorrectly. Please double-check the information given.

To solve this problem, let's start by assigning variables to the unknowns. Let the tens digit be represented by 'x', and the ones digit be represented by 'y'.

We are given the following information:
1) "A number of two digits divided by the sum of the digits the quotient is 7 and the remainder is 6."
This can be represented as (10x + y) / (x + y) = 7 + (6 / (x + y)).

2) "If the digits of the number are interchanged, the resulting number exceeds three times the sum of the digits by 5."
This can be represented as (10y + x) = 3(x + y) + 5.

Now, let's solve these equations simultaneously to find the values of 'x' and 'y'.

From equation 1, we have:
(10x + y) / (x + y) = 7 + (6 / (x + y))

Multiplying through by (x + y):
10x + y = 7(x + y) + 6

Expanding and rearranging:
10x + y = 7x + 7y + 6

Simplifying:
3x - 6y = -6 ----(1)

Similarly, from equation 2, we have:
(10y + x) = 3(x + y) + 5

Expanding and rearranging:
10y + x = 3x + 3y + 5

Simplifying:
-2x + 7y = 5 ----(2)

Now, we have a system of two linear equations with two variables (x and y), which we can solve using any preferred method (such as substitution or elimination).

Using the substitution method, we can rearrange equation (1) to solve for x in terms of y:
3x = 6y - 6
x = 2y - 2

Substituting this value of x into equation (2):
-2(2y - 2) + 7y = 5

Expanding and simplifying:
-4y + 4 + 7y = 5
3y = 1
y = 1/3

Now, substitute this value of y back into x = 2y - 2:
x = 2(1/3) - 2
x = 2/3 - 6/3
x = -4/3

However, the digits of a number cannot be negative or fractional. Therefore, there are no valid two-digit numbers that satisfy the given conditions.

In conclusion, there is no such number that meets the given criteria.