Find the smallest number by which 18000 should be divided so that it becomes a perfect square . - DSSSB - 28 August 2019 (Shift-1)

DSSSB - 28 August 2019 (Shift-1)

Non Nursing Subjects

Hard

Appeared in: DSSSB - 28 August 2019 (Shift-1)

Explanation

The core principle is that a number is a perfect square if all the exponents in its prime factorization are even.
The prime factorization of 18000 is calculated as 2⁴ × 3² × 5³.
In this factorization, the prime factors 2 and 3 have even exponents (4 and 2, respectively), but the prime factor 5 has an odd exponent (3).
To make the exponent of 5 even, the number must be divided by 5¹ (or simply 5).
Dividing 18000 by 5 results in 3600, which is a perfect square (60²), confirming that 5 is the correct smallest divisor.

Option A: Dividing 18000 by 6 (which is 2 × 3) gives 3000. The prime factorization of 3000 is 2³ × 3¹ × 5³, which has three odd exponents. Thus, 3000 is not a perfect square.
Option B: Dividing 18000 by 2 gives 9000. The prime factorization of 9000 is 2³ × 3² × 5³. This result still has odd exponents for the prime factors 2 and 5, so it is not a perfect square.
Option D: Dividing 18000 by 3 gives 6000. The prime factorization of 6000 is 2⁴ × 3¹ × 5³. This result has odd exponents for the prime factors 3 and 5, so it is not a perfect square.

Clinical Relevance

Nursing practice connection: This is primarily an exam-oriented knowledge point with limited direct bedside application, so retain Finding the smallest divisor to make a number a perfect square using prime factorization as background academic context rather than a clinical decision trigger.
This question assesses foundational mathematical aptitude, specifically in number theory, which is a component of the general intelligence and reasoning sections of many competitive examinations, including those for nursing positions.
Strong problem-solving and logical reasoning skills are valuable for nurses in clinical settings for tasks like medication dosage calculations, interpreting lab results, and making quick, evidence-based decisions.
What if the question asked for the smallest number to be multiplied to make it a perfect square? The process is similar. You would still identify the prime factors with odd exponents and multiply by them to make the exponents even. In this case, you would also multiply by 5.

How to Approach the Question

First, identify the core task: find the smallest number to divide 18000 by to get a perfect square.
Recall the key property of perfect squares: in their prime factorization, every prime factor must have an even exponent.
Perform the prime factorization of the number 18000. A good strategy is to break it into smaller, manageable parts like 18 and 1000.
Examine the exponents of each prime factor in the final factorization (2⁴ × 3² × 5³).
Identify any prime factors that have an odd exponent. Here, only the factor 5 has an odd exponent (3).
The smallest number to divide by is the product of all prime factors with odd exponents. In this case, it's just 5.

Concept Tested & Keywords

Concept Tested: Finding the smallest divisor to make a number a perfect square using prime factorization.
Stem keywords: smallest number, 18000, divided, perfect square
Lead-in keywords: Find

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