Given:
- A natural number P leaves a remainder of 11 when divided by K.
- K divides 2P and leaves a remainder of 5.

To find:
The remainder when K divides 4P.

Explanation:
Let's analyze the given information step by step to find a solution.

Remainder when P is divided by K:
When P is divided by K, it leaves a remainder of 11. This can be represented as P ≡ 11 (mod K), where '≡' denotes "congruent" or "equivalent".

Remainder when 2P is divided by K:
When 2P is divided by K, it leaves a remainder of 5. This can be represented as 2P ≡ 5 (mod K).

Using the given information:
We know that P ≡ 11 (mod K) and 2P ≡ 5 (mod K).

Combining the given congruences:
Since 2P ≡ 5 (mod K), we can express 2P as a multiple of K plus the remainder 5. Therefore, 2P = aK + 5, where 'a' is some integer.

Expressing P in terms of K:
Since P ≡ 11 (mod K), we can express P as a multiple of K plus the remainder 11. Therefore, P = bK + 11, where 'b' is some integer.

Substituting the expression for P in terms of K in the expression for 2P:
Substituting P = bK + 11 in the equation 2P = aK + 5, we get:
2(bK + 11) = aK + 5
2bK + 22 = aK + 5
2bK - aK = 5 - 22
(2b - a)K = -17

Determining the value of K:
Since K is a positive integer, the equation (2b - a)K = -17 implies that K must divide -17. The factors of -17 are -17, -1, 1, and 17. Since K is positive, the possible values of K are 1 and 17.

Considering K = 1:
If K = 1, then the remainder when P is divided by 1 would be 0, which contradicts the given information that P leaves a remainder of 11 when divided by K. Therefore, K ≠ 1.

Considering K = 17:
If K = 17, then P ≡ 11 (mod 17) and 2P ≡ 5 (mod 17).

Finding the value of P:
To determine the value of P, we can find a number that satisfies the congruence P ≡ 11 (mod 17). This means P = 17x + 11, where 'x' is some integer. By substituting different