Find the Least Common Multiple (LCM) of 64 and 56.

Introduction / Context:
This question asks for the Least Common Multiple, or LCM, of two numbers, 64 and 56. LCM problems are standard in arithmetic and aptitude exams, as they help test a student's understanding of multiples, prime factorization, and divisibility. The LCM represents the smallest number that both given numbers can divide into exactly, and it is particularly important for solving problems involving synchronization, repeated events, and fraction addition with different denominators.

Given Data / Assumptions:

• First number = 64.
• Second number = 56.
• We need to determine the LCM of these two numbers.
• Both numbers are positive integers.

Concept / Approach:
One of the most reliable ways to find the LCM is to use prime factorization. We express each number as a product of prime powers. Then we form the LCM by taking the highest power of each prime that appears in any of the factorizations. This ensures that the resulting number is divisible by each original number. Using this approach removes guesswork and is especially helpful when numbers share several common prime factors.

Step-by-Step Solution:
Step 1: Prime factorize each number. 64 = 2^6. 56 = 2^3 * 7. Step 2: Identify all distinct primes involved. The primes are 2 and 7. Step 3: Take the highest power of each prime. For 2, highest power is 2^6 (from 64). For 7, highest power is 7^1 (from 56). Step 4: Multiply these to get the LCM. LCM = 2^6 * 7 = 64 * 7 = 448. Therefore, the LCM of 64 and 56 is 448.

Verification / Alternative check:
We can verify the result by checking divisibility. First, 448 / 64 = 7, which is an integer. Second, 448 / 56 = 8, which is also an integer. Thus, 448 is a common multiple of both 64 and 56. To confirm it is the least such positive multiple, consider that any smaller multiple of 64 would be of the form 64 * k with k less than 7, and none of those values are divisible by 56. Similarly, smaller multiples of 56 will not be divisible by 64. So 448 is indeed the least common multiple.

Why Other Options Are Wrong:
488: This number is not divisible by 64, so it cannot be the LCM.
484: 484 is not divisible by 64 and 56 together, hence not a common multiple.
408: 408 is not divisible by 64 without remainder, so it is not suitable.
512: While 512 is a multiple of 64, it is not a multiple of 56; also, it is larger than the correct LCM and therefore not the least common multiple.

Common Pitfalls:
Students sometimes confuse LCM with HCF or try to find the answer by listing common multiples, which is slow and error prone. Another common error is to forget to take the highest power of each prime, especially when one number is a higher power of a prime than the other. Using prime factorization carefully or the relationship with HCF can make these problems quick and reliable to solve.

Final Answer:
The LCM of 64 and 56 is 448.