Find the least common multiple (L.C.M.) of the two numbers 12 and 18.

Introduction:
This is a straightforward L.C.M. question designed to test basic understanding of prime factorisation and the rule for finding the least common multiple of two numbers. Such questions are fundamental in aptitude tests and form the basis for more complex problems.

Given Data / Assumptions:

• The two numbers are 12 and 18.
• We need to compute their L.C.M.

Concept / Approach:
To find the L.C.M. of two numbers using prime factorisation:

• Write each number as a product of prime powers.
• For each prime appearing in either number, take the highest exponent.
• Multiply these prime powers together to get the L.C.M.

Step-by-Step Solution:
Step 1: Factorise 12 and 18. 12 = 2^2 * 3. 18 = 2 * 3^2. Step 2: Collect all distinct primes: 2 and 3. Step 3: For L.C.M., choose the highest power of each prime. For 2: max exponent is 2 (from 12). For 3: max exponent is 2 (from 18). Step 4: L.C.M. = 2^2 * 3^2. Step 5: Compute: 2^2 = 4, 3^2 = 9. Step 6: L.C.M. = 4 * 9 = 36.

Verification / Alternative check:
Check that 36 is divisible by both 12 and 18: 36 ÷ 12 = 3 (integer). 36 ÷ 18 = 2 (integer). Also, there is no smaller positive number that is divisible by both 12 and 18, so 36 is indeed the least common multiple.

Why Other Options Are Wrong:
42: Not divisible by 12, so cannot be the L.C.M. 12: Divisible by 12 but not by 18. 6: Divisible by neither 12 nor 18 as a multiple, it is a common factor, not L.C.M. 18: Divisible by 18 but not by 12.

Common Pitfalls:
A typical mistake is to confuse L.C.M. with H.C.F. and choose the smaller number. Another common error is to multiply the numbers directly (12 * 18 = 216) without reducing by their common factor, resulting in a non-minimal multiple.

Final Answer:
The least common multiple of 12 and 18 is 36.