Find the least common multiple (LCM) of the two numbers 15 and 12.

Introduction:
This is a straightforward question on least common multiple (LCM) of two numbers. Questions like this commonly appear in the HCF and LCM section of aptitude examinations and are meant to test basic factorisation skills.

Given Data / Assumptions:

• Two numbers: 15 and 12
• We want the least positive integer that is exactly divisible by both 15 and 12.

Concept / Approach:
To find the LCM using prime factorisation, we express each number as a product of prime factors. The LCM is obtained by taking each distinct prime with the highest exponent that occurs in any of the numbers. Multiplying these together gives the LCM.

Step-by-Step Solution:
Prime factorise each number: 15 = 3 × 5 12 = 2^2 × 3 Collect all primes involved: 2, 3, and 5 Take the highest power of each prime: For 2: highest power is 2^2 For 3: highest power is 3^1 For 5: highest power is 5^1 LCM = 2^2 × 3 × 5 Calculate step by step: 2^2 = 4 4 × 3 = 12 12 × 5 = 60 So, LCM(15, 12) = 60

Verification / Alternative check:
Check divisibility: 60 ÷ 15 = 4 and 60 ÷ 12 = 5. Both quotients are integers, so 60 is a common multiple. Also, no smaller number than 60 can be simultaneously divisible by both 15 and 12, because any smaller candidate would lack sufficient powers of 2 or 5.

Why Other Options Are Wrong:
3 is a common factor, not a common multiple. 12 is divisible by 3 but not by 15, so it cannot be the LCM. 30 is a common multiple of 15 but not of 12, because 30 ÷ 12 is not an integer. Hence, only 60 satisfies all the conditions for being the least common multiple.

Common Pitfalls:
Students sometimes confuse HCF and LCM and mistakenly select a common factor like 3 or 12 instead of a common multiple. Another typical error is to add or subtract the two numbers rather than use factorisation. Always remember that LCM refers to multiples, not factors.

Final Answer:
The least common multiple of 15 and 12 is 60.