Given Information:
- When a number is divided by 7, the remainder is 3.
- When the resulting quotient is divided by 11, the remainder is 6.

Objective:
To find the values of m and n when the same number is divided by 11 and 7, respectively.

Approach:
1. Let's assume the number as 'x'.
2. According to the given information, we can write the first condition as: x ≡ 3 (mod 7).
This means that x leaves a remainder of 3 when divided by 7.
3. Using this information, we can express x as: x = 7a + 3, where 'a' is an integer.
This equation represents x in terms of the quotient (a) and remainder (3) when divided by 7.
4. Now, let's consider the second condition stated: (7a + 3) ≡ 6 (mod 11).
This means that the quotient of x divided by 7 (which is 'a') leaves a remainder of 6 when divided by 11.
5. Simplifying the equation (7a + 3) ≡ 6 (mod 11), we get: 7a ≡ 3 (mod 11).
This equation represents the relationship between the quotient (a) and the remainder (3) when divided by 11.
6. Now, we need to solve the congruence equation 7a ≡ 3 (mod 11) to find the value of 'a'.
7. We can multiply both sides of the congruence equation by the modular inverse of 7 modulo 11 (if it exists) to isolate 'a'.
8. By calculating the modular inverse of 7 modulo 11, we find that it is 8. Hence, we have 7 * 8 ≡ 1 (mod 11).
Multiplying both sides of the congruence equation by 8, we get: 56a ≡ 24 (mod 11).
9. Simplifying further, we have: a ≡ 2 (mod 11).
This means that the quotient 'a' leaves a remainder of 2 when divided by 11.
10. Now, substituting the value of 'a' in the equation x = 7a + 3, we get: x = 7(2) + 3 = 17.
Hence, the value of x is 17.
11. Finally, we can find the values of m and n.
- When x is divided by 11, the remainder is: 17 % 11 = 6.
Hence, the value of m is 6.
- When the quotient of x divided by 7 is divided by 7, the remainder is: (17/7) % 7 = 3.
Hence, the value of n is 3.

Conclusion:
The values of m and n, when the same number is divided by 11 and 7 respectively, are 6 and 3.