Introduction:
We have been given a positive number that gives a remainder of 8 when divided by 88. We are asked to determine what the remainder will be when this number is divided by 11. To solve this problem, we will use the concepts of modular arithmetic and division properties.

Modular Arithmetic:
Modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" after a certain value called the modulus. In this case, the modulus is 88. The remainder when a number is divided by the modulus is called the residue.

Given Information:
- The number gives a remainder of 8 when divided by 88.

Finding the Number:
To find the number, we can set up an equation using modular arithmetic.

Let the number be represented by N.

N ≡ 8 (mod 88)

This equation means that N is congruent to 8 modulo 88, or in other words, N leaves a remainder of 8 when divided by 88.

Understanding Modulo 11:
To determine the remainder when N is divided by 11, we need to find the residue of N modulo 11. If a number is congruent to 0 modulo 11, it is divisible by 11 without any remainder.

Applying Modular Arithmetic:
We can apply modular arithmetic to the equation N ≡ 8 (mod 88) to find the residue of N modulo 11.

N ≡ 8 (mod 88)
N ≡ 8 (mod 11)

Since 88 is not divisible by 11, we need to find a number that is congruent to 8 modulo 88 and also congruent to a value modulo 11.

Using Common Divisor:
To find a number that satisfies both congruences, we can consider the common divisor of 88 and 11, which is 1.

Applying Chinese Remainder Theorem:
Using the Chinese Remainder Theorem or properties of modular arithmetic, we can find a unique solution for N modulo 11.

Since 88 and 11 are relatively prime (have a common divisor of 1), we can use the Chinese Remainder Theorem to find the unique solution for N modulo 11.

N ≡ 8 (mod 88)
N ≡ 8 (mod 11)

Solving these congruences, we find that the remainder when N is divided by 11 is 8.

Conclusion:
The remainder when the positive number is divided by 11 is 8. This result is obtained by applying modular arithmetic and considering the congruences of the number modulo 88 and modulo 11.

Let the no. is 88+8=96now divide 96 by 11then remainder=8so ans=8