In finding the HCF of two numbers by division method, the quotients are 4, 6 and 8, respectively, and the last divisor is 61. What is the LCM of the two numbers?

1. A

   609753

2. B

   609756

3. C

   609758

4. D

   609761

Appeared in: Bihar NHM CHO-2025

Explanation

• The problem requires working backward through the Euclidean algorithm (division method for HCF) to find the original numbers.
• The last divisor given (61) is the Highest Common Factor (HCF) of the two numbers.
• Using the formula Dividend = (Divisor × Quotient) + Remainder for each step in reverse, the two original numbers are calculated to be 12444 and 2989.
• The Least Common Multiple (LCM) is then found using the fundamental relationship: Product of Numbers = HCF × LCM.
• The final calculation is LCM = (12444 × 2989) / 61, which equals 609756.

Why Other Options Were Wrong

• Option A: This is an incorrect value that would result from a calculation error during the multi-step process, such as an error in multiplication or addition while working backward.
• Option C: This value is close to the correct answer but is incorrect. It likely arises from a minor arithmetic mistake in one of the intermediate steps of the calculation.
• Option D: This is another incorrect value that would be the result of a calculation error, possibly in the final step of finding the LCM or during the backward calculation of the original numbers.

Related Visual

• Visual 1: Flowchart - A flowchart demonstrating the backward calculation process of the Euclidean algorithm. It would start from the last divisor (HCF) and work its way up through the quotients to find the two original numbers, clearly showing how the divisor and remainder of one step become the dividend and divisor of the next.

Clinical Relevance

• Nursing practice connection: This is primarily an exam-oriented knowledge point with limited direct bedside application, so retain HCF by Division Method (Euclidean Algorithm) and its relationship with LCM as background academic context rather than a clinical decision trigger.
• This question tests quantitative aptitude, a non-nursing subject area that is a common component of nursing recruitment examinations to assess general intelligence and problem-solving skills.
• Strong numerical and logical reasoning skills are foundational for many critical nursing tasks, including accurate medication dosage calculation, interpreting vital signs and lab results, and managing patient data.
• What if the last divisor was 1? This would signify that the two original numbers are co-prime (their only common factor is 1). In that case, their LCM would simply be the product of the two numbers.

How to Approach the Question

• Step 1: Identify the given data: the sequence of quotients (4, 6, 8) and the final divisor (61). Recognize that the final divisor in this method is the HCF of the two numbers. So, HCF = 61.
• Step 2: Recall the relationship in the division algorithm: Dividend = (Divisor × Quotient) + Remainder.
• Step 3: Work backward from the last division step, where the remainder is always 0.
• Step 4: For the last step (quotient=8, divisor=61), calculate the dividend: Dividend = (61 × 8) + 0 = 488. This dividend becomes the divisor for the previous step.
• Step 5: For the middle step (quotient=6, divisor=488), the remainder is the previous divisor (61). Calculate the dividend: Dividend = (488 × 6) + 61 = 2989. This is one of the original numbers.
• Step 6: For the first step (quotient=4, divisor=2989), the remainder is the previous divisor (488). Calculate the dividend: Dividend = (2989 × 4) + 488 = 12444. This is the second original number.

Concept Tested & Keywords

• Concept Tested: HCF by Division Method (Euclidean Algorithm) and its relationship with LCM
• Stem keywords: HCF, division method, quotients, last divisor, LCM
• Lead-in keywords: What is

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