The L.C.M. of $22$, $54$, $108$, $135$ and $198$ is

### Concept & Strategy The Lowest Common Multiple (LCM) of a set of numbers is the smallest integer divisible by all numbers in the set. For larger numbers, use **Prime Factorization**. The LCM is the product of the highest power of each unique prime factor present. ### Step-by-Step Solution * **Step 1:** Express each number as a product of its prime factors. * $22 = 2^1 \times 11^1$ * $54 = 2^1 \times 3^3$ * $108 = 2^2 \times 3^3$ * $135 = 3^3 \times 5^1$ * $198 = 2^1 \times 3^2 \times 11^1$ * **Step 2:** Identify all unique prime bases: $2$, $3$, $5$, and $11$. * **Step 3:** Select the highest exponent for each prime base. * Highest power of $2$ is $2^2$ ($4$) * Highest power of $3$ is $3^3$ ($27$) * Highest power of $5$ is $5^1$ ($5$) * Highest power of $11$ is $11^1$ ($11$) * **Step 4:** Multiply these together. $$LCM = 2^2 \times 3^3 \times 5 \times 11$$ $$LCM = 4 \times 27 \times 5 \times 11$$ $$LCM = 108 \times 55 = 5940$$ ### Exam Strategy & Shortcut Use divisibility rules on the options. The LCM must be a multiple of $198$, which is $18 \times 11$. Therefore, the answer must be divisible by $11$. Apply the divisibility rule for $11$ (difference between sum of alternating digits must be $0$ or a multiple of $11$): * (a) $330 \rightarrow (3+0) - 3 = 0$ (Divisible by 11) * (b) $1980 \rightarrow (1+8) - (9+0) = 0$ (Divisible by 11) * (c) $5940 \rightarrow (5+4) - (9+0) = 0$ (Divisible by 11) * (d) $11880 \rightarrow (1+8+0) - (1+8) = 0$ (Divisible by 11) Since all are divisible by $11$, check divisibility by $27$ (from $108$). The sum of digits of $5940$ is $18$, which is divisible by $9$, and $5940 \div 9 = 660$, not divisible by $3$. Wait, let's check $108$. $5940 \div 108 = 55$. Yes, it works. For speed, calculate the $LCM(108, 135) = 540$, then $LCM(540, 198) = 5940$. ### Common Pitfall The continuous long division method is highly prone to arithmetic errors with a large set of numbers like this. Prime factorization is much safer and easier to track. ### Final Answer **Therefore, the correct answer is 5940.**