The product of two numbers is 4107 and the HCF of these numbers is 37.

To find the greater number, we need to find the prime factors of 4107 and identify which factors are common with the HCF.

Step 1: Prime factorization of 4107
4107 can be factorized as follows:
4107 = 3 * 3 * 457

So, the prime factors of 4107 are 3, 3, and 457.

Step 2: Identifying common factors with HCF
Since the HCF is 37, we need to check if any of the prime factors of 4107 are divisible by 37.
37 cannot divide 3 or 457, so the common factor is 1.

Step 3: Finding the greater number
Since the product of the two numbers is 4107 and the HCF is 37, we can express the two numbers as:
Number 1 = HCF * x = 37 * x
Number 2 = HCF * y = 37 * y

The product of the two numbers is given by:
Number 1 * Number 2 = 4107
(37 * x) * (37 * y) = 4107

Simplifying the equation:
37 * 37 * x * y = 4107
37^2 * x * y = 4107

Since 37 is a prime number, it cannot be divided further. So, x and y are the remaining factors.

Step 4: Finding x and y
To find the values of x and y, we can divide 4107 by 37^2:
4107 / (37^2) = 3

So, x * y = 3

Step 5: Finding the greater number
Since the greater number is either x or y, we need to identify which factor is greater.

Since x * y = 3 and x and y are positive integers, the possible combinations are (x, y) = (1, 3) or (3, 1).

Number 1 = 37 * x = 37 * 1 = 37
Number 2 = 37 * y = 37 * 3 = 111

So, the greater number is 111.

The greater number is 111.