Find the LCM (Lowest Common Multiple) of 15, 21 and 30

Appeared in: RRB Staff Nurse Kolkata-2015

Explanation

The Lowest Common Multiple (LCM) is the smallest number that is a multiple of all given numbers.
To find the LCM of 15, 21, and 30, we use prime factorization: 15 = 3 x 5, 21 = 3 x 7, and 30 = 2 x 3 x 5.
The LCM is the product of the highest powers of all prime factors present in the numbers.
The unique prime factors are 2, 3, 5, and 7.
The highest power of 2 is 2¹, of 3 is 3¹, of 5 is 5¹, and of 7 is 7¹.
Multiplying these gives the LCM: 2 x 3 x 5 x 7 = 210.

Option B: The number 3 is the Greatest Common Divisor (GCD) or Highest Common Factor (HCF) of 15, 21, and 30. It is the largest number that divides all three numbers, not the smallest number they all divide into.
Option C: While 630 is a common multiple of 15, 21, and 30 (15 x 42 = 630, 21 x 30 = 630, 30 x 21 = 630), it is not the lowest common multiple. The LCM must be the smallest such number.
Option D: The number 105 is a multiple of 15 (15 x 7) and 21 (21 x 5), but it is not a multiple of 30. Since the LCM must be divisible by all the numbers, 105 is incorrect.

Visual 1: Flowchart: A flowchart illustrating the step-by-step process of finding the LCM using the prime factorization method. It would show breaking down each number, collecting the highest powers of primes, and multiplying them to get the final answer.

Clinical Relevance

Nursing practice connection: This is primarily an exam-oriented knowledge point with limited direct bedside application, so retain Lowest Common Multiple (LCM) as background academic context rather than a clinical decision trigger.
While not a clinical skill, quantitative aptitude is a component of many nursing entrance and recruitment examinations.
Mastery of basic mathematical concepts like LCM and HCF is essential for scoring well in the general aptitude sections of these competitive exams.
What if the question asked for the HCF? The Highest Common Factor (HCF) or Greatest Common Divisor (GCD) is the largest number that divides all the given numbers. The prime factors are 15=3x5, 21=3x7, 30=2x3x5. The only common prime factor is 3, so the HCF would be 3.

How to Approach the Question

First, understand the question. It asks for the Lowest Common Multiple (LCM) of three numbers: 15, 21, and 30.
Recall the definition of LCM: the smallest positive integer that is divisible by all the given numbers.
Choose an appropriate method. Prime factorization is a reliable method for finding the LCM.
Step 1: Find the prime factors of each number (15 = 3 x 5; 21 = 3 x 7; 30 = 2 x 3 x 5).
Step 2: Identify all unique prime factors (2, 3, 5, 7) and take the highest power of each that appears in any factorization (2¹, 3¹, 5¹, 7¹).
Step 3: Multiply these highest powers together (2 x 3 x 5 x 7) to calculate the LCM. The result is 210.

Concept Tested & Keywords

Q11usWMvlUb_P9RZ1ZMiQp

Practise the full RRB Staff Nurse Kolkata-2015

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