What is the Least Common Multiple (LCM) of the two numbers 120 and 450?

Introduction / Context:
This question requires finding the Least Common Multiple, or LCM, of 120 and 450. LCM problems are essential in quantitative aptitude because they relate to combining cycles, synchronizing events, and adding or comparing fractions with different denominators. Understanding how to compute LCM efficiently helps you avoid time consuming trial and error methods in exams.

Given Data / Assumptions:

• First number = 120.
• Second number = 450.
• We need to find their LCM.
• Both numbers are positive integers.

Concept / Approach:
The LCM of two numbers can be found using prime factorization or using the formula: LCM(a, b) = (a * b) / HCF(a, b). Often it is efficient to first find the Highest Common Factor (HCF) using prime factorization or the Euclidean algorithm, and then apply this formula. This is especially useful when the numbers are fairly large but share a significant common factor, as it reduces the amount of direct multiplication and manual checking needed.

Step-by-Step Solution:
Step 1: Prime factorize both numbers. 120 = 2^3 * 3 * 5. 450 = 2 * 3^2 * 5^2. Step 2: Determine the HCF. HCF uses the smallest power of each common prime. Common primes are 2, 3, and 5. For 2, smallest power is 2^1. For 3, smallest power is 3^1. For 5, smallest power is 5^1. So HCF = 2 * 3 * 5 = 30. Step 3: Use the LCM formula. LCM = (120 * 450) / 30. Step 4: Simplify the product with division. First, 450 / 30 = 15. Now LCM = 120 * 15 = 1800. Therefore, the LCM of 120 and 450 is 1800.

Verification / Alternative check:
We can verify by prime factorization directly for the LCM. From the factorizations: 120 = 2^3 * 3 * 5, 450 = 2 * 3^2 * 5^2. LCM uses the highest powers of each prime: 2^3 (from 120), 3^2 (from 450), 5^2 (from 450). Thus, LCM = 2^3 * 3^2 * 5^2 = 8 * 9 * 25 = 72 * 25 = 1800. This matches the value found using the formula, confirming that 1800 is correct.

Why Other Options Are Wrong:
2400: Although 2400 is a multiple of both 120 and 450, it is not the least such multiple because we have already found a smaller common multiple, 1800.
3600 and 4800: These are also common multiples of 120 and 450 but are larger than the correct LCM and therefore not minimal.
900: This number is a multiple of 450 but not of 120, since 900 / 120 is not an integer, so it cannot be the LCM.

Common Pitfalls:
Some students multiply the numbers directly and forget to divide by the HCF, leading to very large and incorrect answers. Others might miscalculate the HCF or confuse highest and lowest prime powers when handling factorizations. Using the systematic approach of prime factorization plus the formula for LCM minimizes these errors and makes it easier to verify the result.

Final Answer:
The LCM of 120 and 450 is 1800.