Problem:
Find the smallest number by which 980 must be multiplied so that the product is a perfect square.

Solution:
To find the smallest number by which 980 must be multiplied to obtain a perfect square, we need to factorize 980 and determine the prime factors and their powers.

Factorization of 980:
To factorize 980, we can start by dividing it by the smallest prime number, which is 2.

980 ÷ 2 = 490

So, the factorization of 980 is: 2 × 2 × 5 × 7 × 7 = 2^2 × 5 × 7^2

Prime factors and their powers:
From the factorization, we can see that the prime factors of 980 are 2, 5, and 7, with powers 2, 1, and 2 respectively.

Finding the smallest number for a perfect square:
To obtain a perfect square, we need to make the powers of all prime factors even. In this case, the power of 2 is already even (2^2 = 4), but the powers of 5 and 7 are odd (5^1 = 5 and 7^2 = 49).

To make the powers even, we need to multiply 980 by the smallest prime factors with odd powers. In this case, it is 5.

980 × 5 = 4900

Now, let's check if the product 4900 is a perfect square.

Checking if 4900 is a perfect square:
To check if 4900 is a perfect square, we can find its square root.

√4900 = 70

Since the square root of 4900 is an integer (70), we can conclude that 4900 is a perfect square.

Therefore, the smallest number by which 980 must be multiplied to obtain a perfect square is 5, making the product 4900, which is a perfect square.

Hence, the correct answer is option 'B' (5).