Let's denote the two co-prime numbers by $a$ and $b$.

Step 1: Define the multiples

According to the problem, one multiple of $a$ and $b$ is $20$ less than $100$, and the next multiple is $20$ more than $100$.

This can be written as: $ab = 100 - 20 = 80$ and $\text{Next multiple} = ab + 20 + 20 = 120$

Step 2: Factorization

We know $ab = 80$. We will list pairs of factors of 80 to find $a$ and $b$ such that they are co-prime (i.e., their greatest common divisor (GCD) is 1).

The factor pairs of 80 are:

• $1 \times 80$
• $2 \times 40$
• $4 \times 20$
• $5 \times 16$
• $8 \times 10$

Step 3: Check for Co-prime Pairs

We now check each of these pairs to see if they are co-prime.

• $\gcd(1, 80) = 1$ (co-prime)
• $\gcd(2, 40) = 2$ (not co-prime)
• $\gcd(4, 20) = 4$ (not co-prime)
• $\gcd(5, 16) = 1$ (co-prime)
• $\gcd(8, 10) = 2$ (not co-prime)

Step 4: Verify the Next Multiple

From the list, the co-prime pairs are $(1, 80)$ and $(5, 16)$.

We now check the next multiple:

• For $(1, 80)$, the next multiple is $80 + 80 = 160$ (this does not satisfy the problem's condition).
• For $(5, 16)$, the next multiple is $80 + 40 = 120$ (this satisfies the condition).

Conclusion:

The co-prime numbers are $\boxed{5 \text{ and } 16}$.

Would you like more details or have any questions?

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Here are 8 related questions you might find interesting:

1. What are the properties of co-prime numbers?
2. How can we find the GCD of two numbers efficiently?
3. What is the significance of the Least Common Multiple (LCM) in problems like this?
4. Can two consecutive numbers be co-prime?
5. What is the smallest possible product of two co-prime numbers?
6. How does the Euclidean algorithm help in finding co-prime pairs?
7. Can you provide another example of finding co-prime numbers with similar conditions?
8. How would the problem change if the numbers were not co-prime?

Tip: Remember, two numbers are co-prime if their GCD is 1, meaning they have no common factors other than 1.