State fundamental theorem of Arithmetic. Is it possible that HCF and LCM of two numbers be 24 and 540 respectively. Justify your answer.

Answer

### Step-by-Step Solution

**Step 1: State the Fundamental Theorem of Arithmetic.**
The Fundamental Theorem of Arithmetic states that every integer greater than 1 can be expressed uniquely as a product of prime factors, up to the order of the factors. This means that any integer can be broken down into prime numbers in only one way.

**Step 2: Understand the relationship between HCF and LCM.**
The relationship between the Highest Common Factor (HCF) and the Least Common Multiple (LCM) of two numbers \(a\) and \(b\) is given by the formula:
\[ \text{HCF} \times \text{LCM} = a \times b \]
This means that the product of the HCF and LCM of two numbers is equal to the product of the numbers themselves.

**Step 3: Given values of HCF and LCM.**
In this case, we are given:
- HCF = 24
- LCM = 540

**Step 4: Calculate the product of HCF and LCM.**
Using the relationship stated in Step 2, we calculate:
\[ \text{Product} = \text{HCF} \times \text{LCM} = 24 \times 540 \]

**Step 5: Calculate the product.**
Now, we perform the multiplication:
\[ 24 \times 540 = 12960 \]

**Step 6: Analyze the implications of the product.**
If \(a\) and \(b\) are the two numbers, then:
\[ a \times b = 12960 \]
This means that the product of the two numbers must equal 12960.

**Step 7: Check for possible values of HCF and LCM.**
Now, we need to check if it is possible for two numbers to have an HCF of 24 and an LCM of 540.

**Step 8: Find the prime factorization of HCF and LCM.**
- The prime factorization of 24 is \(2^3 \times 3^1\).
- The prime factorization of 540 is \(2^2 \times 3^3 \times 5^1\).

**Step 9: Determine the prime factorization of the two numbers.**
Let the two numbers be \(a\) and \(b\). The HCF (24) indicates that the common prime factors must include \(2^3\) and \(3^1\). The LCM (540) indicates that the maximum powers of the prime factors must include \(2^2\), \(3^3\), and \(5^1\).

**Step 10: Check consistency of prime factors.**
Since the HCF must be less than or equal to both numbers, and the LCM must be greater than or equal to both numbers, we can see that:
- The HCF cannot exceed the maximum powers of the prime factors in the LCM.
- However, the HCF (24) has \(2^3\), which exceeds the maximum power of \(2\) in the LCM (which is \(2^2\)).

**Step 11: Conclusion.**
Thus, it is not possible for two numbers to have an HCF of 24 and an LCM of 540, because the conditions for the prime factors contradict each other.

### Final Answer
No, it is not possible for two numbers to have an HCF of 24 and an LCM of 540.

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