What is the least common multiple (L.C.M.) of the numbers 22, 54, 108, 135 and 198?

Introduction / Context:
Finding the least common multiple (L.C.M.) of a set of numbers is important in many arithmetic and algebra problems, especially those involving synchronization of cycles, finding common periods, or adding and subtracting fractions. In this question, we must determine the L.C.M. of five numbers: 22, 54, 108, 135 and 198.

Given Data / Assumptions:

• Numbers: 22, 54, 108, 135 and 198.
• We need the smallest positive number that is divisible by each of these.
• All numbers are positive integers.

Concept / Approach:
To find the L.C.M.:

• Write the prime factorization of each number.
• For each distinct prime, take the highest power that appears in any factorization.
• Multiply these highest powers together to get the L.C.M.

This approach guarantees the smallest number that contains all required prime factors in sufficient powers.

Step-by-Step Solution:
Step 1: Factor each number into primes. 22 = 2 * 11. 54 = 2 * 3^3 (since 54 = 2 * 27 = 2 * 3^3). 108 = 2^2 * 3^3 (since 108 = 4 * 27). 135 = 3^3 * 5 (since 135 = 27 * 5). 198 = 2 * 3^2 * 11 (since 198 = 2 * 99 = 2 * 9 * 11). Step 2: Identify all distinct primes: 2, 3, 5 and 11. Step 3: For each prime, find the highest power appearing in any number. For 2: highest power is 2^2 (from 108). For 3: highest power is 3^3 (from 54, 108 and 135). For 5: highest power is 5^1 (from 135). For 11: highest power is 11^1 (from 22 and 198). Step 4: Multiply these highest powers: L.C.M. = 2^2 * 3^3 * 5 * 11. Step 5: Compute step by step: 2^2 = 4, 3^3 = 27, so 4 * 27 = 108. Step 6: Multiply 108 by 5: 108 * 5 = 540. Step 7: Multiply 540 by 11: 540 * 11 = 5940.

Verification / Alternative Check:
Check divisibility:

• 5940 ÷ 22 = 270 exactly.
• 5940 ÷ 54 = 110 exactly.
• 5940 ÷ 108 = 55 exactly.
• 5940 ÷ 135 = 44 exactly.
• 5940 ÷ 198 = 30 exactly.

Since 5940 is divisible by each number without remainder, it is a common multiple. Because we used the highest prime powers method, it is also the least such common multiple.

Why Other Options Are Wrong:
330 and 1980 are too small; they are not divisible by all the given numbers, especially 108 and 135.
11880 is a multiple of 5940, but it is larger and therefore not the least common multiple.
Thus, 5940 is the smallest positive integer divisible by all five numbers.

Common Pitfalls:
A common mistake is to forget one of the primes or to choose a lower power than required, which leads to a number that is not a common multiple. Others may create a multiple that works but is not the least one. Sticking to prime factorization and highest powers prevents these errors.

Final Answer:
The L.C.M. of 22, 54, 108, 135 and 198 is 5940.