Let's solve this step-by-step:
Given:
- The greatest common factor (gcd) of two whole numbers is 5.
- The least common multiple (lcm) of these same two numbers is 100.
To find the two numbers \( a \) and \( b \), we need to understand the relationship between gcd and lcm:
[tex]\[
\text{gcd}(a, b) \times \text{lcm}(a, b) = a \times b
\][/tex]
Given:
[tex]\[
\text{gcd}(a, b) = 5
\][/tex]
[tex]\[
\text{lcm}(a, b) = 100
\][/tex]
We substitute these values into the relationship formula:
[tex]\[
5 \times 100 = a \times b
\][/tex]
This simplifies to:
[tex]\[
a \times b = 500
\][/tex]
We need to find pairs of \( a \) and \( b \) whose product is 500 and which also satisfy that their gcd is 5.
Let's list pairs whose product is 500:
[tex]\[
(1, 500), (2, 250), (4, 125), (5, 100), (10, 50), (20, 25), (25, 20), (50, 10), (100, 5), and (125, 4)
\][/tex]
Now, we need to check the gcd of each pair to find pairs with gcd equal to 5:
- For pair (5, 100): GCD(5, 100) = 5
- For pair (10, 50): GCD(10, 50) = 10
- For pair (20, 25): GCD(20, 25) = 5
Checking other pairs quickly shows none have gcd of 5.
Thus, our valid pairs are:
[tex]\[
(5, 100) \text{ and } (20, 25)
\][/tex]
Since one number must be 5 (as per options, there can be repetitions), among our choices:
[tex]\[
\text{Only } (5, 20) \text{ and } (20, 5) fit.
\][/tex]
Thus each valid set of numbers solutions are:
[tex]\[
5 \text{ and } 20 \text{ (or equivalently } 20 \text{ and } 5\text{).}
\][/tex]
Thus, the correct answer from the choices given is:
[tex]\[
\boxed{5 \text{ and } 20}
\][/tex]
So, the correct option is:
B. 5 and 20
