What is the least common multiple (LCM) of the integers 120 and 450?

Introduction:
This problem focuses on finding the least common multiple (LCM) of two numbers, 120 and 450. Such questions are directly useful in solving problems related to repeating events, scheduling and synchronization in aptitude tests.

Given Data / Assumptions:

• First number = 120.
• Second number = 450.
• We need their least common multiple.

Concept / Approach:
We use prime factorization to determine the LCM by taking each prime factor with the highest power that appears in either number.
120 = 2^3 * 3 * 5450 = 2 * 3^2 * 5^2Then we select the maximum power of each prime: 2^3, 3^2 and 5^2, and multiply them to get the LCM.

Step-by-Step Solution:
Step 1: Factorize 120.120 = 2^3 * 3 * 5Step 2: Factorize 450.450 = 2 * 3^2 * 5^2Step 3: Choose highest powers of each prime.For 2: max exponent = 3 ⇒ 2^3For 3: max exponent = 2 ⇒ 3^2For 5: max exponent = 2 ⇒ 5^2Step 4: Form the LCM.LCM = 2^3 * 3^2 * 5^2Step 5: Compute numerically.2^3 = 83^2 = 95^2 = 258 * 9 = 7272 * 25 = 1800

Verification / Alternative check:
Check that 1800 is divisible by both numbers. 1800 ÷ 120 = 15 and 1800 ÷ 450 = 4. Both are integers, so 1800 is a common multiple. Because we used the largest exponents of all relevant primes minimally, no smaller positive number can be a common multiple, confirming 1800 as the LCM.

Why Other Options Are Wrong:
2400, 3600, 4800: These are multiples of 1800 or other larger numbers, so although they may be common multiples, they are not the least.
900: 900 is divisible by 450, but 900 ÷ 120 = 7.5, which is not an integer, so it is not even a common multiple.

Common Pitfalls:
Candidates may directly multiply 120 and 450 to get 54000 and mistakenly think that is the LCM. Prime factorization or using LCM = (a * b) / HCF ensures much smaller and correct answers for such problems.

Final Answer:
The least common multiple of 120 and 450 is 1800.