What is the least common multiple (LCM) of the two integers 57 and 93?

Introduction:
This problem asks you to find the least common multiple (LCM) of two numbers, 57 and 93. Such LCM questions are simple when you apply prime factorization or use the relationship between product, HCF and LCM.

Given Data / Assumptions:

• First number = 57.
• Second number = 93.
• We need their least common multiple.

Concept / Approach:
We can use prime factorization:
57 = 3 * 1993 = 3 * 31The LCM is obtained by taking each prime that appears in either factorization with the highest power that appears. Alternatively, for two numbers, we can also use:
LCM(a, b) = (a * b) / HCF(a, b)

Step-by-Step Solution:
Step 1: Prime factorize.57 = 3 * 1993 = 3 * 31Step 2: Identify distinct primes.Primes involved are 3, 19 and 31.Step 3: LCM is product of these primes with exponent 1.LCM = 3 * 19 * 31Step 4: Compute stepwise.3 * 19 = 5757 * 31 = 1767

Verification / Alternative check:
Check divisibility: 1767 ÷ 57 = 31, an integer, and 1767 ÷ 93 = 19, also an integer. Thus 1767 is a common multiple of 57 and 93. There is no smaller positive integer with this property because we used the minimal prime factors needed, so 1767 is indeed the least common multiple.

Why Other Options Are Wrong:
1567, 1576, 1919: These are not multiples of both 57 and 93. Dividing them by 57 or 93 does not give an integer.
114: While 114 is a common multiple of 57 and 93 (114 ÷ 57 = 2, 114 ÷ 93 is not an integer), so even this is not a common multiple of both. Hence only 1767 qualifies.

Common Pitfalls:
One common mistake is to assume that LCM must be close to or smaller than the product a * b, without actually doing prime factorization. Another is to miscalculate the multiplication 57 * 31. Taking it step by step avoids such arithmetic slips.

Final Answer:
The least common multiple of 57 and 93 is 1767.