The H.C.F. of $2^2 \times 3^3 \times 5^5$, $2^3 \times 3^2 \times 5^2 \times 7$ and $2^4 \times 3^4 \times 5 \times 7^2 \times 11$ is
Aptitude
HCF and LCM
Difficulty: Medium
Choose an option
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A$2^2 \times 3^2 \times 5$
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B$2^2 \times 3^2 \times 5 \times 7 \times 11$
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C$2^4 \times 3^4 \times 5^5$
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D$2^4 \times 3^4 \times 5^5 \times 7 \times 11$
Answer
Correct Answer: $2^2 \times 3^2 \times 5$
Explanation
### Concept & Formula
To find the Highest Common Factor (HCF) of numbers already given in their prime factorized form, you do not need to expand or multiply them out.
**The Rule for HCF of Prime Expressions:**
Select only the prime bases that are **common** to all the given numbers, and raise each to its **lowest** power present across the expressions.
### Step-by-Step Solution
* **Given:** We have three numbers expressed as products of primes:
Number 1: $2^2 \times 3^3 \times 5^5$
Number 2: $2^3 \times 3^2 \times 5^2 \times 7$
Number 3: $2^4 \times 3^4 \times 5 \times 7^2 \times 11$
* **Calculation / Deduction:**
1. **Identify Common Bases:** Look at the prime bases ($2, 3, 5, 7, 11$). Only the bases $2, 3,$ and $5$ appear in all three expressions. (We ignore $7$ and $11$ because they are missing from the first expression).
2. **Find the Lowest Power of 2:** The powers of $2$ are $2^2, 2^3, 2^4$. The lowest is $2^2$.
3. **Find the Lowest Power of 3:** The powers of $3$ are $3^3, 3^2, 3^4$. The lowest is $3^2$.
4. **Find the Lowest Power of 5:** The powers of $5$ are $5^5, 5^2, 5^1$ (since $5$ is $5^1$). The lowest is $5^1$ or just $5$.
5. Multiply these lowest powers together to get the HCF:
$$HCF = 2^2 \times 3^2 \times 5$$
### Exam Strategy & Shortcut
**Rapid Elimination:**
You can solve this by scanning the options instantly.
Since the first expression lacks $7$ and $11$, any option containing $7$ or $11$ is immediately incorrect. This quickly eliminates options (b) and (d).
Next, look at the base $5$. The third expression only has $5^1$. Therefore, the HCF cannot have a power of $5$ greater than $1$. This eliminates option (c) which contains $5^5$.
Option (a) is the only survivor.
### Common Pitfall
Confusing the rules for HCF and LCM. Students often mistakenly select the *highest* powers or include all distinct prime factors (which calculates the LCM, not HCF). Remember: HCF equals Common bases only, Lowest powers.
### Final Answer
**Therefore, the correct answer is $2^2 \times 3^2 \times 5$.**