Problem:
By which least number should 5000 be divided so that it becomes a perfect square?

Solution:
To find the least number by which 5000 should be divided to become a perfect square, we need to analyze the prime factors of 5000.

Prime Factorization of 5000:
To find the prime factors of 5000, we can divide it successively by prime numbers until the quotient becomes 1.

5000 ÷ 2 = 2500
2500 ÷ 2 = 1250
1250 ÷ 2 = 625
625 ÷ 5 = 125
125 ÷ 5 = 25
25 ÷ 5 = 5
5 ÷ 5 = 1

Therefore, the prime factorization of 5000 is 2 × 2 × 2 × 5 × 5 × 5 × 5.

Perfect Square:
A number is a perfect square if all its prime factors occur in pairs. In other words, each prime factor should have an even exponent.

In the prime factorization of 5000, we have three 2's and four 5's. To make 5000 a perfect square, we need to pair these prime factors.

Pairing Prime Factors:
Since we have three 2's, we can pair them to get one 2 with an exponent of 2 (2^2).

For the remaining prime factors, we have four 5's. We can pair two of them to get one 5 with an exponent of 2 (5^2).

Now, we have 2^2 × 5^2 = 4 × 25 = 100.

Therefore, the least number by which 5000 should be divided to become a perfect square is 100.

Checking the Options:
Now let's check the options given in the question.

a) 2: 5000 ÷ 2 = 2500 (not a perfect square)
b) 5: 5000 ÷ 5 = 1000 (not a perfect square)
c) 10: 5000 ÷ 10 = 500 (not a perfect square)
d) 25: 5000 ÷ 25 = 200 (not a perfect square)

Since none of the options except option 'A' (2) result in a perfect square, the correct answer is option 'A'.