To solve this problem, we can use the concept of congruence and modular arithmetic.

Let's assume the number we are trying to find is represented by 'x'.

When x is divided by 387, the remainder obtained is 48. We can write this as:

x ≡ 48 (mod 387)

This means that x is congruent to 48 modulo 387. In other words, x and 48 leave the same remainder when divided by 387.

Now, we are asked to find the remainder when x is divided by 43. We can write this as:

x ≡ r (mod 43)

Our goal is to find the value of 'r'.

To solve this, we need to find a number that satisfies both congruences. In other words, we need to find a number that leaves a remainder of 48 when divided by 387 and also leaves the same remainder when divided by 43.

To find such a number, we can use the Chinese Remainder Theorem (CRT). The CRT states that if we have a system of congruences with pairwise coprime moduli, then there exists a unique solution modulo the product of the moduli.

In this case, the moduli 387 and 43 are not coprime, since their greatest common divisor (GCD) is 43. However, we can still use a modified version of the CRT.

We can rewrite the congruence x ≡ 48 (mod 387) as:

x ≡ 387k + 48 (mod 387)

where 'k' is an integer.

Now, we substitute this expression into the congruence x ≡ r (mod 43):

387k + 48 ≡ r (mod 43)

We can simplify this congruence by reducing the coefficients modulo 43:

7k + 5 ≡ r (mod 43)

Now, we can try different values for 'k' and find a value that satisfies the congruence. We notice that when 'k' is 6, the congruence holds:

7(6) + 5 ≡ 47 ≡ 4 (mod 43)

Therefore, the remainder when x is divided by 43 is 4, which corresponds to option (C) in the given choices.