Introduction:
Co-prime numbers are numbers that have no common factors other than 1. In other words, when two numbers are co-prime, their greatest common divisor (GCD) is 1. In this question, we need to find the numbers up to 100 that are co-prime to 19.

Method:
To find the numbers up to 100 that are co-prime to 19, we need to eliminate all the numbers that have a common factor with 19.

Step 1: Find the prime factorization of 19.
19 is a prime number, so its prime factorization is 19 x 1.

Step 2: Eliminate all the numbers that have 19 as a factor.
This means we need to eliminate all multiples of 19 from the list of numbers up to 100. This includes 19, 38, 57, 76, and 95.

Step 3: Find the GCD of each remaining number with 19.
To be co-prime, the GCD of each remaining number with 19 must be 1. We can use the Euclidean algorithm to find the GCD of each number with 19. If the GCD is 1, the number is co-prime to 19.

Step 4: List the co-prime numbers.
The numbers up to 100 that are co-prime to 19 are: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 96, 97, 98, 99, and 100.

Conclusion:
To summarize, the numbers up to 100 that are co-prime to 19 are the numbers that have no common factors with 19. We found these numbers by eliminating all multiples of 19 from the list of numbers up to 100 and then checking the GCD of each remaining number with 19. The final list of co-prime numbers includes all numbers that have a GCD of 1 with 19.