What is the least common multiple (LCM) of the two integers 64 and 56?

Introduction:
This is a straightforward LCM calculation involving two numbers. It tests your understanding of prime factorization and how to use it to calculate the least common multiple quickly and accurately.

Given Data / Assumptions:

• First number = 64.
• Second number = 56.
• We are asked to find their least common multiple (LCM).

Concept / Approach:
The LCM of two integers can be found by using prime factorization. We express each number as a product of prime powers and then take the highest power of each prime that appears in any factorization.
LCM = product of primes raised to maximum exponents appearing in either number

Step-by-Step Solution:
Step 1: Prime factorize each number.64 = 2^656 = 2^3 * 7Step 2: Take the highest exponent for each prime.For prime 2: max exponent is 6 (from 64).For prime 7: max exponent is 1 (from 56).Step 3: Form the LCM.LCM = 2^6 * 7Step 4: Compute the value.2^6 = 6464 * 7 = 448

Verification / Alternative check:
Check that 448 is divisible by both numbers. 448 ÷ 64 = 7, and 448 ÷ 56 = 8. Both quotients are integers, so 448 is a common multiple. Because it is constructed from prime factors with minimal exponents that cover both numbers, it is the least such multiple.

Why Other Options Are Wrong:
488, 484, 408, 512: None of these are the true LCM. For example, 512 is a multiple of 64 but not of 56. The others either fail divisibility by 64 or 56 or both. Only 448 satisfies divisibility by both numbers and is the smallest such number.

Common Pitfalls:
Learners sometimes mistakenly multiply the two numbers directly (64 * 56) instead of factoring them, which gives 3584, a common multiple but not the least. Another error is miscalculating the prime exponents, such as using 2^3 instead of 2^6. Careful factorization prevents these errors.

Final Answer:
The least common multiple of 64 and 56 is 448.