Find the LCM (Least Common Multiple) of 54 and 60. Choose the correct value of LCM(54, 60).

Introduction / Context:This question tests LCM calculation for two numbers using prime factorization or the gcd method. The LCM is the smallest number that both inputs divide exactly. It is a fundamental concept in arithmetic and number theory, especially useful in scheduling and fraction problems.

Given Data / Assumptions:

• Numbers: 54 and 60
• Goal: LCM(54, 60)

Concept / Approach:We use prime factorization. Express each number in prime powers, then:LCM = product of each prime raised to its highest exponent across the numbersAn alternative method for verification is the identity:LCM(x, y) * gcd(x, y) = x * y

Step-by-Step Solution:Step 1: Factorize 54.54 = 2 * 27 = 2 * 3^3.Step 2: Factorize 60.60 = 2^2 * 3 * 5.Step 3: Take maximum exponents of each prime:2: max exponent = 2^23: max exponent = 3^35: max exponent = 5^1Step 4: LCM = 2^2 * 3^3 * 5 = 4 * 27 * 5 = 540.

Verification / Alternative check:Compute gcd(54, 60). The gcd is 6. Then:LCM = (54 * 60) / gcd(54, 60) = 3240 / 6 = 540This matches the prime factorization method, confirming that 540 is correct.

Why Other Options Are Wrong:600: 600 / 54 is not an integer, so 600 is not a common multiple of both numbers.360: 360 is not divisible by 54.270: Not divisible by 60.720: Although it is a common multiple, it is larger than the least common multiple, which is 540.

Common Pitfalls:Using HCF instead of LCM when reading the question quickly.Multiplying the numbers directly without dividing by their gcd, which gives a non-minimal common multiple.Missing a prime factor or using a lower exponent rather than the highest exponent across the factorizations.

Final Answer:540