What is the Least Common Multiple (LCM) of the two numbers 57 and 93?

Introduction / Context:
This question requires you to find the Least Common Multiple, or LCM, of the numbers 57 and 93. Such questions test your ability to factorize numbers into primes and to use those factorizations to compute the smallest common multiple. Being efficient with these operations is very helpful when solving problems involving repeated patterns, time schedules, and fraction operations, where LCM naturally appears.

Given Data / Assumptions:

• First number = 57.
• Second number = 93.
• We need to determine their LCM.
• Both numbers are positive integers.

Concept / Approach:
The prime factorization method is the most reliable for LCM. We factor each number into primes, then build the LCM by taking the highest power of each prime that appears in any factorization. Alternatively, another efficient approach is to use the relationship: LCM(a, b) = (a * b) / HCF(a, b), provided we can find the HCF of the two numbers quickly. In this case, both methods are easy to apply, because the numbers are not extremely large and share some obvious factors.

Step-by-Step Solution:
Step 1: Perform prime factorization. 57 = 3 * 19. 93 = 3 * 31. Step 2: Identify distinct primes from both factorizations. The primes are 3, 19, and 31. Step 3: Take the highest power of each prime. All primes appear to the first power, so we use 3^1, 19^1, and 31^1. Step 4: Multiply these prime powers to obtain the LCM. LCM = 3 * 19 * 31. Compute: 3 * 19 = 57. Then 57 * 31 = 1767. Therefore, the LCM of 57 and 93 is 1767.

Verification / Alternative check:
We can also use the relationship with HCF. First, find HCF(57, 93). Since both numbers are divisible by 3, divide: 57 / 3 = 19 and 93 / 3 = 31. The resulting numbers, 19 and 31, are both prime and share no further common factors. Thus, HCF(57, 93) = 3. Now use the formula: LCM = (57 * 93) / 3. Compute 57 * 93 = 5301. Then 5301 / 3 = 1767, which matches the earlier result. This confirms that 1767 is correct.

Why Other Options Are Wrong:
1567, 1576, and 1919: These numbers do not match the prime factor structure 3 * 19 * 31, and they are not simultaneously divisible by both 57 and 93.
114: While 114 is a multiple of 57 and 93 shares some factors with it, 93 does not divide 114 exactly, so it cannot serve as their LCM.

Common Pitfalls:
Sometimes students guess the LCM by multiplying the numbers directly without dividing by the HCF, which gives the product rather than the least common multiple. Others may incorrectly compute the HCF or miss a prime factor. Using prime factorization or the formula involving HCF keeps the process clear and minimizes errors, especially when dealing with products of large primes.

Final Answer:
The LCM of 57 and 93 is 1767.