Problem:
Find the smallest number by which 557568 must be divided so that it becomes a perfect square.

Solution:
To find the smallest number by which 557568 must be divided to become a perfect square, we need to first factorize 557568.

Factorization of 557568:
To factorize 557568, we can start by finding its prime factors.

Step 1: Divide 557568 by 2 (smallest prime number) until it is no longer divisible by 2.
557568 ÷ 2 = 278784
278784 ÷ 2 = 139392
139392 ÷ 2 = 69696
69696 ÷ 2 = 34848
34848 ÷ 2 = 17424
17424 ÷ 2 = 8712
8712 ÷ 2 = 4356
4356 ÷ 2 = 2178
2178 ÷ 2 = 1089 (odd number, so we stop dividing by 2)

Step 2: Divide the resulting odd number (1089) by the next prime number (3) until it is no longer divisible by 3.
1089 ÷ 3 = 363
363 ÷ 3 = 121 (odd number, so we stop dividing by 3)

Step 3: Divide the resulting odd number (121) by the next prime number (5) until it is no longer divisible by 5.
121 ÷ 5 = 24.2 (not divisible by 5)

Step 4: Divide the resulting number (24.2) by the next prime number (7) until it is no longer divisible by 7.
24.2 ÷ 7 = 3.457 (not divisible by 7)

Since 3.457 is not a whole number, we can conclude that the prime factorization of 557568 is 2^7 * 3^2 * 7^2.

Perfect Square:
In order for a number to be a perfect square, all its prime factors must have an even exponent.

In the prime factorization of 557568, we have:
2^7 * 3^2 * 7^2

To make it a perfect square, we need to divide the odd exponents by 2.

2^7 * 3^2 * 7^2 can be written as:
2^6 * 3^1 * 7^1

Therefore, the smallest number by which 557568 must be divided to become a perfect square is:
2^6 * 3^1 * 7^1 = 64 * 3 * 7 = 1344

Hence, the correct answer is option 'D' - 2.