" Which of the following must be odd? I. The sum of 5 consecutive integers II. The sum of 14 consecutive integers III. The product of 11 consecutive integers (A) Only I (B) Only II (C) Only III (D) Both I and II"

Question

"  Which of the following must be odd?  I. The sum of 5 consecutive integers II. The sum of 14 consecutive integers III. The product of 11 consecutive integers (A)     Only I (B)     Only II (C)     Only III (D)     Both I and II"

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Analyze Option I: Let's analyze option I: The sum of 5 consecutive integers. We can represent 5 consecutive integers as x, x+1, x+2, x+3, x+4. The sum of these integers is x + (x+1) + (x+2) + (x+3) + (x+4). Simplify the sum: 5x + (1+2+3+4) = 5x + 10. Since 5x is always a multiple of 5, and 10 is even, the sum 5x + 10 will always be even, regardless of the value of x. Therefore, the sum of 5 consecutive integers is not necessarily odd.Analyze Option II: Let's analyze option II: The sum of 14 consecutive integers. We can represent 14 consecutive integers as x, x+1, x+2, ..., x+13. The sum of these integers is x + (x+1) + (x+2) + ... + (x+13). Simplify the sum: 14x + (1+2+3+...+13) = 14x + 91. Since 14x is always a multiple of 14, and 91 is odd, the sum 14x + 91 will always be odd, because an even number plus an odd number is odd. Therefore, the sum of 14 consecutive integers is necessarily odd.Analyze Option III: Let's analyze option III: The product of 11 consecutive integers. We can represent 11 consecutive integers as x, x+1, x+2, ..., x+10. The product of these integers is x * (x+1) * (x+2) * ... * (x+10). Since the sequence contains 11 numbers, at least one of them must be even (because every second integer is even). The product of an even number with any other numbers is always even. Therefore, the product of 11 consecutive integers is not necessarily odd; it is always even.Final Conclusion: Based on the analysis, only the sum of 14 consecutive integers must be odd. The correct answer is (B) Only II.

Which of the following must be odd? $\newline$ I. The sum of $5$ consecutive integers $\newline$ II. The sum of $14$ consecutive integers $\newline$ III. The product of $11$ consecutive integers $\newline$ (A) Only I $\newline$ (B) Only II $\newline$ (C) Only III $\newline$ (D) Both I and II

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Which of the following must be odd? $\newline$ I. The sum of $5$ consecutive integers $\newline$ II. The sum of $14$ consecutive integers $\newline$ III. The product of $11$ consecutive integers $\newline$ (A) Only I $\newline$ (B) Only II $\newline$ (C) Only III $\newline$ (D) Both I and II