Find the least common multiple (LCM) of the two numbers 72 and 84.

Introduction:
This question tests your ability to compute the least common multiple (LCM) of two numbers using prime factorisation. The numbers 72 and 84 are typical values used in aptitude questions to practice LCM calculations.

Given Data / Assumptions:

• Two numbers: 72 and 84
• We must find their LCM.

Concept / Approach:
The LCM of two numbers is the smallest positive integer that is a multiple of both numbers. Using prime factorisation, we express each number as a product of primes and then take each prime with the highest exponent found in either number. Multiplying those prime powers gives the LCM.

Step-by-Step Solution:
Prime factorise the numbers: 72 = 2^3 × 3^2 84 = 2^2 × 3 × 7 List all primes involved: 2, 3, and 7. Select the highest powers: For 2: highest power is 2^3 For 3: highest power is 3^2 For 7: highest power is 7^1 So LCM = 2^3 × 3^2 × 7 Compute step by step: 2^3 = 8 3^2 = 9 8 × 9 = 72 72 × 7 = 504 Therefore LCM(72, 84) = 504

Verification / Alternative check:
Check divisibility: 504 ÷ 72 = 7 and 504 ÷ 84 = 6. Both results are integers, so 504 is indeed a common multiple of 72 and 84. Any smaller candidate would not contain enough factors of 2, 3, or 7 to be divisible by both numbers.

Why Other Options Are Wrong:
252 is divisible by 84 but not by 72. 360 and 420 are also not divisible by both 72 and 84 simultaneously. Hence none of these can be the LCM, even though they are smaller than 504. Only 504 satisfies the definition of the least common multiple.

Common Pitfalls:
Some students may mistakenly multiply the two numbers directly, leading to 6048, which is a common multiple but not the least. Others may forget to take the highest powers of primes and instead use lower powers, which leads to a number that is not divisible by one of the originals.

Final Answer:
The least common multiple of 72 and 84 is 504.