Find the largest number which can exactly divide 513, 783 and 1107.

Appeared in: RRB Staff Nurse Jaipur-6th June 2018

Explanation

The question asks for the 'largest number which can exactly divide' a set of numbers, which is the definition of the Highest Common Factor (HCF), also known as the Greatest Common Divisor (GCD).
The prime factorization method is used to find the HCF. We break down each number into its prime factors.
513 = 3 × 3 × 3 × 19.
783 = 3 × 3 × 3 × 29.
1107 = 3 × 3 × 3 × 41.
The common factors in all three numbers are 3 × 3 × 3, which equals 27. Therefore, 27 is the HCF.

Option B: 25 is not a factor of any of the given numbers. A number must end in 0 or 5 to be divisible by 5, and none of the given numbers do.
Option C: 21 has prime factors 3 and 7. While all the numbers are divisible by 3, they are not divisible by 7 (e.g., 513 ÷ 7 = 73.28). For a number to be a factor, all its prime components must also be factors.
Option D: 23 is a prime number and is not a factor of any of the given numbers (e.g., 513 ÷ 23 = 22.30).

Visual 1: Infographic: A step-by-step visual guide showing the prime factorization tree for 513, 783, and 1107, highlighting the common factors (3x3x3) to derive the HCF of 27.
Visual 2: Flowchart: A flowchart illustrating the Euclidean Algorithm to find the HCF of multiple numbers, using the numbers from the question as an example.

Clinical Relevance

Nursing practice connection: This is primarily an exam-oriented knowledge point with limited direct bedside application, so retain Highest Common Factor (HCF) as background academic context rather than a clinical decision trigger.
This question tests basic arithmetic and logical reasoning, which are foundational skills for nurses.
Accurate calculation is critical in nursing for tasks like medication dosage calculation, IV drip rate regulation, and interpreting patient data, where errors can have serious consequences.
What if the question asked for the 'smallest number that is exactly divisible by' 9, 12, and 15? The approach would change to finding the Least Common Multiple (LCM), not the HCF. The LCM would be 180.

How to Approach the Question

First, understand the question's terminology. 'Largest number which can exactly divide' is a key phrase that translates to finding the Highest Common Factor (HCF).
Choose a method to find the HCF. The two common methods are Prime Factorization and the Euclidean Algorithm.
For the Prime Factorization method, break down each number (513, 783, 1107) into its prime factors.
Identify the prime factors that are common to all the numbers.
Multiply these common prime factors together to get the HCF.
Compare your result with the given options to find the correct answer.

Concept Tested & Keywords

QnOA1J_chMtBSgbF3bO1CN

Practise the full RRB Staff Nurse Jaipur-6th June 2018

Attempt every question from this paper in a timed mock, then review the full solution for each one.