Find the least common multiple (L.C.M.) of the two numbers 54 and 60.

Introduction:
This problem reinforces the method of finding the least common multiple (L.C.M.) using prime factorisation. Working with two composite numbers like 54 and 60 helps you practise identifying prime powers and using them correctly to compute the L.C.M.

Given Data / Assumptions:

• The numbers are 54 and 60.
• We need to find their L.C.M.

Concept / Approach:
For two numbers a and b:

• Prime factorise each number.
• For each prime that appears in either factorisation, take the highest exponent.
• Multiply those prime powers together to get the L.C.M.

This ensures the L.C.M. is the smallest number divisible by both a and b.

Step-by-Step Solution:
Step 1: Factorise 54. 54 = 6 * 9 = 2 * 3^3. Step 2: Factorise 60. 60 = 6 * 10 = 2^2 * 3 * 5. Step 3: Gather all primes: 2, 3, and 5. Step 4: Take maximum exponents. For 2: max exponent is 2 (from 60). For 3: max exponent is 3 (from 54). For 5: max exponent is 1 (from 60). Step 5: L.C.M. = 2^2 * 3^3 * 5. Step 6: Compute: 2^2 = 4; 3^3 = 27. Step 7: 4 * 27 = 108. Step 8: 108 * 5 = 540.

Verification / Alternative check:
Check divisibility: 540 ÷ 54 = 10 (integer). 540 ÷ 60 = 9 (integer). Since 540 is divisible by both numbers and is generated from maximum prime exponents, it is indeed their least common multiple.

Why Other Options Are Wrong:
600: Divisible by 60 but not by 54, since 600 ÷ 54 is not an integer. 60 and 54: Each is one of the numbers, not a common multiple of both. 108: Divisible by 54 but not by 60, so it cannot be the L.C.M.

Common Pitfalls:
Many students mistakenly multiply the numbers directly instead of using prime factorisation, getting a much larger multiple. Others may forget to include a prime (like 5 here) or use the smaller exponent instead of the larger one when computing the L.C.M.

Final Answer:
The least common multiple of 54 and 60 is 540.