H.C.F. of $4 \times 27 \times 3125$, $8 \times 9 \times 25 \times 7$ & $16 \times 81 \times 5 \times 11 \times 49$ is

### Concept & Strategy Before applying the HCF rule for exponential expressions, all numbers must be converted into their **prime** factorized forms. The given numbers contain composite bases (like $4, 27, 25, 16, 81$, etc.) which must be broken down first. Once converted to prime bases, the HCF is the product of the lowest powers of the common prime factors. ### Step-by-Step Solution * **Given:** Expression 1: $4 \times 27 \times 3125$ Expression 2: $8 \times 9 \times 25 \times 7$ Expression 3: $16 \times 81 \times 5 \times 11 \times 49$ * **Calculation / Deduction:** 1. **Convert Expression 1 to Primes:** $4 = 2^2$ $27 = 3^3$ $3125 = 5^5$ So, Exp 1 = $2^2 \times 3^3 \times 5^5$ 2. **Convert Expression 2 to Primes:** $8 = 2^3$, $9 = 3^2$, $25 = 5^2$, $7 = 7^1$ So, Exp 2 = $2^3 \times 3^2 \times 5^2 \times 7^1$ 3. **Convert Expression 3 to Primes:** $16 = 2^4$, $81 = 3^4$, $5 = 5^1$, $11 = 11^1$, $49 = 7^2$ So, Exp 3 = $2^4 \times 3^4 \times 5^1 \times 11^1 \times 7^2$ 4. **Find Common Primes:** The bases present in *all three* expressions are $2, 3$, and $5$. (Ignore $7$ and $11$ as they are not in Exp 1). 5. **Extract Lowest Powers:** Lowest power of $2$ is $2^2$ (from Exp 1) Lowest power of $3$ is $3^2$ (from Exp 2) Lowest power of $5$ is $5^1$ (from Exp 3) 6. **Calculate HCF:** $$HCF = 2^2 \times 3^2 \times 5^1$$ $$HCF = 4 \times 9 \times 5 = 180$$ ### Exam Strategy & Shortcut **Targeting the Smallest Component:** Look for the smallest factor among the composite groupings. For base 5: The third expression has a standalone "$5$". This means the highest power of 5 in the HCF can only be $5^1 = 5$. For base 2: The first expression has "$4$". The highest power of 2 in the HCF can only be $4$. For base 3: The second expression has "$9$". The highest power of 3 in the HCF can only be $9$. Multiply these limiting factors directly: $4 \times 9 \times 5 = 180$. You don't need to calculate $3125$ or $81$ at all. ### Common Pitfall Applying the HCF rule directly to the composite numbers without prime-factorizing them first. If you just look for "common numbers" in the raw expressions, you will completely miss the hidden prime factors shared across the composite bases. ### Final Answer **Therefore, the correct answer is $180$.**