The H.C.F. of $2^4 \times 3^2 \times 5^3 \times 7$, $2^3 \times 3^3 \times 5^2 \times 7^2$ and $3 \times 5 \times 7 \times 11$ is

### Concept & Strategy To find the Highest Common Factor (HCF) of numbers provided in their prime factorized form, you only need to look at the prime bases. The rule is: Identify the prime bases that are common to **all** expressions, and take the **lowest** power of each common base present. Multiply these together to get the HCF. ### Step-by-Step Solution * **Given:** Expression 1: $2^4 \times 3^2 \times 5^3 \times 7^1$ Expression 2: $2^3 \times 3^3 \times 5^2 \times 7^2$ Expression 3: $3^1 \times 5^1 \times 7^1 \times 11^1$ * **Calculation / Deduction:** 1. **Identify Common Bases:** Look at the prime bases across all three expressions. The bases $3, 5$, and $7$ appear in every expression. The base $2$ is missing from the third expression, and $11$ is missing from the first two. 2. **Lowest Power of 3:** Comparing $3^2, 3^3,$ and $3^1$, the lowest is $3^1$ (or simply $3$). 3. **Lowest Power of 5:** Comparing $5^3, 5^2,$ and $5^1$, the lowest is $5^1$ (or simply $5$). 4. **Lowest Power of 7:** Comparing $7^1, 7^2,$ and $7^1$, the lowest is $7^1$ (or simply $7$). 5. **Calculate Final HCF:** Multiply these common lowest powers together: $$HCF = 3 \times 5 \times 7$$ $$HCF = 15 \times 7 = 105$$ ### Exam Strategy & Shortcut **Rapid Elimination:** You can spot the answer in under 10 seconds. The third expression is just $3 \times 5 \times 7 \times 11$. Since 2 is missing, the HCF cannot be even. Options (c) $2310$ and (d) $27720$ are even numbers, so eliminate them instantly. Furthermore, 11 is not present in the first expression, so 11 cannot be in the HCF. Option (b) $1155$ is divisible by 11 ($1+5 = 6$, $1+5 = 6$). This leaves only $105$ as the correct choice. ### Common Pitfall The most frequent mistake is getting confused with the LCM rule and picking the *highest* powers, or blindly multiplying all distinct prime factors together. Always remember: HCF means "Common" bases and "Lowest" powers. ### Final Answer **Therefore, the correct answer is $105$.**