Solution:

We know that any number which is divisible by 33 is also divisible by 3 and 11. So, we need to check the divisibility of the given expression with 3 and 11.

Divisibility by 3:

The sum of the digits of the number should be divisible by 3 for the entire number to be divisible by 3.

1044 → 1 + 0 + 4 + 4 = 9

1047 → 1 + 0 + 4 + 7 = 12

1050 → 1 + 0 + 5 + 0 = 6

1053 → 1 + 0 + 5 + 3 = 9

The sum of the digits of all the numbers is not divisible by 3. Hence, the given expression is not divisible by 3.

Divisibility by 11:

The difference between the sum of the alternate digits of the number should be either 0 or a multiple of 11 for the entire number to be divisible by 11.

1044 → (1 + 4) - (0 + 4) = 1

1047 → (1 + 7) - (0 + 4) = 4

1050 → (1 + 5) - (0 + 0) = 6

1053 → (1 + 3) - (0 + 5) = -1

The difference of the alternate digits of the numbers is not 0 or a multiple of 11. Hence, the given expression is not divisible by 11.

Therefore, the given expression is not divisible by both 3 and 11.

Using Chinese Remainder Theorem:

We can use the Chinese Remainder Theorem to find the remainder when the expression is divided by 3 and 11.

Remainder when divided by 3:

1044 * 1047 * 1050 * 1053 ≡ 0 * 0 * 0 * 0 ≡ 0 (mod 3)

Remainder when divided by 11:

1044 ≡ 6 (mod 11)

1047 ≡ 9 (mod 11)

1050 ≡ 0 (mod 11)

1053 ≡ 3 (mod 11)

1044 * 1047 * 1050 * 1053 ≡ 6 * 9 * 0 * 3 ≡ 0 (mod 11)

Now, we need to find a number which is congruent to 0 modulo 3 and 0 modulo 11.

Using the Chinese Remainder Theorem, we can find such a number.

Let N = 3 * 11 = 33

Let a1 = 11, a2 = 3

Let b1 = 0, b2 = 0

Since gcd(a1, a2) = 1, we can find two integers c1 and c2 such that c1 * a1 + c2 * a2 = 1.

We can use the Extended Euclidean Algorithm to find c1 and c2.

11 * 3 + (-1) * 33 = 0

c1 = 3, c2 = -1

Therefore, the solution to the congruences is given by:

x ≡ b1 * c2 * a2 + b2 * c1