The product of two numbers is 4107. If the H.C.F. of these numbers is 37, what is the greater of the two numbers?

Introduction / Context:
This is a direct application of the relationship between the product of two numbers and their highest common factor (H.C.F.) when you are asked to determine the individual numbers. It is a standard type of question in quantitative aptitude, and knowing how to factor correctly is key.

Given Data / Assumptions:

• The product of the two numbers is 4107.
• Their H.C.F. is 37.
• We must find the greater of the two numbers.

Concept / Approach:
If two numbers have H.C.F. 37, we can express them as 37a and 37b, where a and b are co-prime positive integers. Then their product is (37a) * (37b) = 37^2 * ab. Since the product is given, we can solve for ab and then find co-prime pairs (a, b) whose product matches this value. This will lead us to the actual numbers, from which we choose the larger.

Step-by-Step Solution:
Step 1: Let the numbers be 37a and 37b, with H.C.F.(a, b) = 1.Step 2: Product = 37^2 * ab.Step 3: Given product is 4107, so 37^2 * ab = 4107.Step 4: Calculate 37^2 = 1369.Step 5: So 1369 * ab = 4107.Step 6: Divide to find ab: ab = 4107 / 1369 = 3.Step 7: The co-prime factor pairs of 3 are (1, 3) and (3, 1).Step 8: The corresponding numbers are 37 * 1 = 37 and 37 * 3 = 111.Step 9: Of these, the greater number is 111.

Verification / Alternative check:
Check product: 37 * 111 = 4107, which matches the given product.Check H.C.F.: gcd(37, 111) is 37 because 111 = 3 * 37. Both conditions are satisfied, confirming that the numbers are 37 and 111 and the greater one is 111.

Why Other Options Are Wrong:
Options a (101), b (107), and d (185) either do not produce the correct product with a compatible partner or do not have H.C.F. 37 with any other integer such that the product is 4107. For example, 101 is prime and 4107 / 101 is not an integer, so it cannot be one of the numbers.

Common Pitfalls:
Students may forget to represent the numbers in the form 37a and 37b and instead try random factorizations of 4107. Others may miscalculate 37^2 or the division 4107 / 1369. Writing the steps cleanly and doing the arithmetic carefully makes this question very manageable.

Final Answer:
The greater of the two numbers is 111.