Analysis:
The given information states that the product of two numbers is 4107 and their highest common factor (HCF) is 37. We are required to find the greater number among the two.

Solution:

Step 1: Prime Factorization of 4107
- To find the prime factorization of 4107, we need to express it as a product of prime numbers.
- 4107 = 3 * 1369
- 1369 = 37 * 37
- Therefore, the prime factorization of 4107 is 3 * 37 * 37.

Step 2: Identifying the Numbers
- Since the HCF of the two numbers is 37, both numbers must have 37 as a common factor.
- Therefore, the two numbers can be expressed as 37 * x and 37 * y, where x and y are the remaining factors after removing 37.
- As the product of the two numbers is 4107, we have 37 * x * 37 * y = 4107.
- Simplifying the equation, we get x * y = 3.
- The possible pairs of x and y that satisfy this condition are (1, 3) and (3, 1).

Step 3: Finding the Greater Number
- The greater number among the two can be calculated by multiplying 37 with the larger factor.
- Therefore, the greater number is 37 * 3 = 111.
Thus, the greater number among the two numbers is 111.