Numbers Between 11 and 19

The set of integers between 11 and 19, inclusive, is: {11, 12, 13, 14, 15, 16, 17, 18, 19}. There are a total of 9 numbers in this set.

Total Pairs Calculation

The total number of possible pairs that can be formed from 9 distinct numbers is calculated using the combination formula $C(n, 2) = \frac{n(n-1)}{2}$, where $n$ is the number of items.

For $n=9$, the total number of pairs is:

$ C(9, 2) = \frac{9 \times (9-1)}{2} = \frac{9 \times 8}{2} = \frac{72}{2} = 36 $

So, there are 36 possible pairs in total.

Non-Coprime Pairs Identification

Two numbers are coprime if their greatest common divisor (GCD) is 1. We need to find pairs whose GCD is greater than 1 (non-coprime).

Let's list the non-coprime pairs (where GCD > 1) from the set {11, 12, 13, 14, 15, 16, 17, 18, 19}:

• Pairs with a common factor of 2 (even numbers): (12, 14), (12, 16), (12, 18), (14, 16), (14, 18), (16, 18)
• Pairs with a common factor of 3: (12, 15), (12, 18), (15, 18)
• Other common factors (like 4, 6) result in pairs already listed above (e.g., GCD(12, 16) = 4, GCD(12, 18) = 6).

Combining and removing duplicates, the unique non-coprime pairs are:

• (12, 14)
• (12, 15)
• (12, 16)
• (12, 18)
• (14, 16)
• (14, 18)
• (15, 18)
• (16, 18)

There are 8 non-coprime pairs.

Coprime Pairs Calculation

To find the number of coprime pairs, subtract the number of non-coprime pairs from the total number of pairs.

Number of coprime pairs = Total pairs - Non-coprime pairs

Number of coprime pairs = $36 - 8 = 28$.