Find the LCD (Least Common Denominator) of 12 and 18. (Note: For positive integers, the LCD is the same as the LCM of the denominators.)

Introduction / Context:
The LCD (Least Common Denominator) of two fractions with denominators 12 and 18 is simply the LCM of 12 and 18. The LCD is important when adding or comparing fractions, because it is the smallest number that both denominators divide into exactly.

Given Data / Assumptions:

• Denominators: 12 and 18
• LCD(12,18) = LCM(12,18)

Concept / Approach:
Use prime factorization to find the LCM. Express each number as a product of prime powers, then take the highest power of each prime that appears in either factorization:
LCM = product of primes with maximum exponents

Step-by-Step Solution:
Step 1: Factorize 12.12 = 2^2 * 3Step 2: Factorize 18.18 = 2 * 3^2Step 3: Combine primes using maximum exponents:Prime 2: max exponent = 2^2Prime 3: max exponent = 3^2Step 4: LCM = 2^2 * 3^2 = 4 * 9 = 36.

Verification / Alternative check:
36 ÷ 12 = 3 and 36 ÷ 18 = 2, both integers, so 36 is a common multiple. Any smaller number, such as 18 or 24, fails to be divisible by both 12 and 18, so 36 is indeed the least common denominator.

Why Other Options Are Wrong:
18: Divisible by 18 but not by 12.42: Not divisible by 12.12: Not divisible by 18.72: Is a common multiple, but it is not the least common multiple.

Common Pitfalls:
Confusing LCD with HCF.Multiplying 12 and 18 directly without reducing by gcd, which gives a non-minimal common multiple.Missing the correct exponent when writing prime factors (for example, forgetting that 18 contains 3^2).

Final Answer:
36