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

Appeared in: HSSC Staff Nurse - 2015

Explanation

The fundamental principle is that if the H.C.F. of two numbers is 'H', the numbers can be expressed as Hx and Hy, where x and y are co-prime.
Given H.C.F. = 37, the numbers are 37x and 37y.
Their product is (37x)(37y) = 4107, which simplifies to 1369xy = 4107, and further to xy = 3.
Since x and y must be co-prime integers, the only possible pair is (1, 3).
The numbers are therefore 37 * 1 = 37 and 37 * 3 = 111.
The question asks for the greater number, which is 111.

Option A: 101 is not a multiple of the H.C.F., which is 37. Any number in the pair must be divisible by their H.C.F.
Option B: 107 is not a multiple of the H.C.F., which is 37. Both numbers must be divisible by 37.
Option D: 185 is a multiple of 37 (37 × 5). If one number were 185, the other would be 4107 / 185 = 22.2, which is not an integer. This violates the condition that we are dealing with two integers.

Clinical Relevance

Nursing practice connection: This is primarily an exam-oriented knowledge point with limited direct bedside application, so retain This question tests the understanding of the relationship between the Highest Common Factor (H.C.F.) and the product of two numbers as background academic context rather than a clinical decision trigger.
This is a non-clinical question designed to test quantitative aptitude, a skill often included in the general aptitude section of nursing entrance examinations.
Proficiency in basic mathematics is essential for various nursing calculations, such as drug dosage, IV drip rates, and interpreting patient data charts, although this specific concept (H.C.F.) is less directly applied.
What if the H.C.F. was 3 and the product was 108? The numbers would be 3x and 3y. (3x)(3y) = 108 -> 9xy = 108 -> xy = 12. Co-prime pairs for 12 are (1, 12) and (3, 4). This would lead to two possible sets of numbers: (3, 36) or (9, 12).

How to Approach the Question

First, identify the key information provided: the product of two numbers (4107) and their H.C.F. (37).
Recall the core property of H.C.F.: If the H.C.F. of two numbers is 'H', then the numbers can be written as 'Hx' and 'Hy', where 'x' and 'y' are co-prime (have no common factors other than 1).
Apply this property to the problem: Let the numbers be 37x and 37y.
Set up an equation using the given product: (37x) * (37y) = 4107.
Solve this equation for the product 'xy'. You will find xy = 3.
Determine the co-prime integer factors of the result (3). The only pair is 1 and 3.

Concept Tested & Keywords

Concept Tested: This question tests the understanding of the relationship between the Highest Common Factor (H.C.F.) and the product of two numbers.
Stem keywords: Product of two numbers, H.C.F, greater number
Lead-in keywords: then the greater number is?

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