Find the LCM (Least Common Multiple) of the three numbers 80, 85, and 90. Choose the correct LCM value.

Introduction / Context:
This problem checks your understanding of LCM using prime factorization. The LCM of several integers is the smallest positive integer that is divisible by each of them. The standard method is to break each number into prime factors and then take each prime to its highest power that appears in any of the numbers.

Given Data / Assumptions:

• Numbers: 80, 85, 90
• Goal: LCM(80, 85, 90)

Concept / Approach:
Using prime factorization, write each number in the form of prime powers. Then:
LCM = product of all primes appearing in any number, each raised to its maximum exponent
This guarantees that the result is divisible by all the numbers and is as small as possible.

Step-by-Step Solution:
Step 1: Factorize 80.80 = 2^4 * 5Step 2: Factorize 85.85 = 5 * 17Step 3: Factorize 90.90 = 2 * 3^2 * 5Step 4: List the primes and pick maximum exponents:2: max power = 2^43: max power = 3^25: max power = 5^117: max power = 17^1Step 5: LCM = 2^4 * 3^2 * 5 * 17 = 16 * 9 * 5 * 17.16 * 9 = 144, 144 * 5 = 720, 720 * 17 = 12240.

Verification / Alternative check:
Check divisibility: 12240 ÷ 80 = 153, 12240 ÷ 85 = 144, 12240 ÷ 90 = 136. All quotients are integers, so 12240 is a common multiple. Because we used the prime-power method, this common multiple is guaranteed to be the least.

Why Other Options Are Wrong:
11440: Does not include 3^2, so it is not divisible by 90.12880: Contains extra factors and is larger than necessary, so it is not the least common multiple.15300: Also a common multiple but not the smallest one.10200: Not divisible by 85, because it lacks the factor 17.

Common Pitfalls:
Forgetting to include the prime 17 from 85 in the LCM.Using addition instead of multiplication when combining prime powers.Confusing LCM with HCF, which uses minimum exponents instead of maximum ones.

Final Answer:
12240