Consider the following statements :<br>1. The sum of the cubes of three consecutive natural numbers is divisible by 9.<br>2. Every even power of every odd number (\gt 1) when divided by 8 gives 1 as remainder.<br>Which of the above statements is/are correct?
- A. 1 only
- B. 2 only
- C. Both 1 and 2 ✓
- D. Neither 1 nor 2
Correct Answer: C. Both 1 and 2
Explanation
Statement 1: Let the numbers be n-1, n, n+1. Sum of cubes = 3n(n^2+2). One of n or n^2+2 is always a multiple of 3, ensuring the product is divisible by 9. Statement 2: (2k+1)^2 = 4k(k+1) + 1, which is always 8m + 1. Thus, (2k+1)^{2m} also leaves a remainder of 1. Both are correct.
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