The given question can be solved using the concept of divisibility and prime factorization. Let's break down the solution into steps:

Step 1: Prime factorization of 500
To find the prime factorization of 500, we divide it by the smallest prime number, which is 2. Repeatedly dividing by 2, we get:

500 ÷ 2 = 250
250 ÷ 2 = 125
125 ÷ 5 = 25
25 ÷ 5 = 5
5 ÷ 5 = 1

So, the prime factorization of 500 is 2 × 2 × 5 × 5 × 5, which can be written as 2^2 × 5^3.

Step 2: Prime factorization of 40
Similarly, for 40:

40 ÷ 2 = 20
20 ÷ 2 = 10
10 ÷ 2 = 5
5 ÷ 5 = 1

So, the prime factorization of 40 is 2^3 × 5.

Step 3: Comparing prime factors
Now, let's compare the prime factors of 500 and 40:

Prime factors of 500: 2^2 × 5^3
Prime factors of 40: 2^3 × 5

From the comparison, we can see that the highest power for the common prime factor '2' is '2' (from 500), and the highest power for the common prime factor '5' is '1' (from 40).

Step 4: Finding the common factor
To find the common factor, we take the product of the common prime factors with their highest powers:

Common factor = 2^2 × 5^1 = 4 × 5 = 20

Step 5: Checking the options
Now, let's check the given options:

a) 1 - The common factor is not 1, so option 'A' is incorrect.
b) 2 - The common factor is not 2, so option 'B' is incorrect.
c) 1 2 - The common factor is not 1 or 2, so option 'C' is incorrect.
d) 1 2 - The common factor is 20, so option 'D' is correct.

Therefore, the correct answer is option 'A'.

In conclusion, the correct answer is option 'A' because the common factor of 500 and 40 is 20, not 1.