Let \(x = n(n+1)(n+2)\), where \(n\) is an even natural number. Which of the following statements is/are correct? I. \(x\) is always divisible by 48. II. \(x^2\) is always divisible by 144. Select the answer using the code given below.
To solve the question, we need to analyze the expression \(x = n(n+1)(n+2)\) where \(n\) is an even natural number, and check the divisibility of \(x\) and \(x^2\) by 48 and 144 respectively.
Step 1: Understanding the Expression
Since \(n\) is even, let \(n = 2k\) where \(k\) is an integer. Thus:
\(x = n(n + 1)(n + 2) = 2k(2k + 1)(2k + 2)\)
Rewriting \((2k + 2)\) as \(2(k + 1)\), we have:
\(x = 2k(2k + 1) \cdot 2(k + 1) = 4k(k+1)(2k+1)\)
Step 2: Determining Divisibility by 48
Divisibility by 48 requires the expression to have factors of 24 and 3.
- We have at least four factors of 2 (from \(4k(k+1)\)), satisfying the power of 2 requirement.
- However, \(k\), \(k+1\), and \(2k+1\) do not necessarily provide a factor of 3. For a concrete example, if \(k = 2\), then \(2k + 1 = 5\) which is not divisible by 3.
Thus, Statement I is incorrect: \(x\) is not always divisible by 48.
Step 3: Determining Divisibility by 144
Divisibility by 144 requires \(x^2\) to be divisible by 144, which means:
\(x^2 = (4k(k+1)(2k+1))^2\) should be divisible by 144 = 24 ├Ч 32.
- We already have \(x\) divisible by 16, hence \(x^2\) is divisible by 162 = 256, providing more than sufficient factors of 2.
- For divisibility by 9, at least one of \(k\), \(k+1\), or \(2k+1\) will be divisible by 3, ensuring that \(x^2\) will be divisible by 9.
Therefore, Statement II is correct: \(x^2\) is always divisible by 144.
Conclusion
The correct answer is:
II only
Thus, the statement that \(x^2\) is always divisible by 144 is correct, while the statement regarding divisibility of \(x\) by 48 is incorrect.
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