Find the least common multiple (L.C.M.) of the three numbers 80, 85, and 90.

Introduction:
Finding the least common multiple (L.C.M.) of several numbers is a standard aptitude skill. This question uses moderately sized numbers to test your comfort with prime factorisation and the rule for computing L.C.M. from prime powers.

Given Data / Assumptions:

• The three numbers are 80, 85, and 90.
• We are required to find their L.C.M.

Concept / Approach:
To find the L.C.M. of several numbers:

• Express each number as a product of prime powers.
• For each prime that appears in any factorisation, take the highest exponent.
• Multiply all these prime powers together to obtain the L.C.M.

Step-by-Step Solution:
Step 1: Factorise each number. 80 = 8 * 10 = 2^4 * 5. 85 = 5 * 17 = 5 * 17. 90 = 9 * 10 = 2 * 3^2 * 5. Step 2: Collect all distinct primes: 2, 3, 5, and 17. Step 3: Choose the highest exponent of each prime across the three numbers. For 2: max exponent is 4 (from 80). For 3: max exponent is 2 (from 90). For 5: max exponent is 1 (all contain at most 5^1). For 17: exponent is 1 (from 85). Step 4: L.C.M. = 2^4 * 3^2 * 5 * 17. Step 5: Compute stepwise: 2^4 = 16; 3^2 = 9. Step 6: 16 * 9 = 144. Step 7: 144 * 5 = 720. Step 8: 720 * 17 = 12240.

Verification / Alternative check:
Confirm that 12240 is divisible by each number: 12240 ÷ 80 = 153 (integer). 12240 ÷ 85 = 144 (integer). 12240 ÷ 90 = 136 (integer). Since 12240 is divisible by all three numbers and is obtained using maximum prime powers, it is indeed their least common multiple.

Why Other Options Are Wrong:
11440, 11998, 12880, 8160: None of these numbers are obtained using the correct prime power combination, and some are not even divisible by all three numbers. For example, 11440 is not divisible by 85, and 8160 is not divisible by 85 either.

Common Pitfalls:
A common mistake is to multiply the numbers directly instead of using prime factors, which leads to unnecessarily large and incorrect results. Another error is to use minimum exponents of primes (which corresponds to H.C.F., not L.C.M.).

Final Answer:
The least common multiple of 80, 85, and 90 is 12240.