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

View step-by-step help

Home

Math Problems

Algebra 1

Consecutive integer problems

Full solution

Q. 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

\newline

(D) Both I and II

Analyze Option I: Let's analyze option I: The sum of $5$ consecutive integers. $\newline$ We can represent $5$ consecutive integers as $x, x+1, x+2, x+3, x+4$ . $\newline$ The sum of these integers is $x + (x+1) + (x+2) + (x+3) + (x+4)$ . $\newline$ Simplify the sum: $5x + (1+2+3+4) = 5x + 10$ . $\newline$ 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 $x$ . $\newline$ 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. $\newline$ We can represent $14$ consecutive integers as $x, x+1, x+2, ..., x+13$ . $\newline$ The sum of these integers is $x + (x+1) + (x+2) + ... + (x+13)$ . $\newline$ Simplify the sum: $14x + (1+2+3+...+13) = 14x + 91$ . $\newline$ 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. $\newline$ Therefore, the sum of $14$ consecutive integers is necessarily odd.
Analyze Option III: Let's analyze option III: The product of $11$ consecutive integers. $\newline$ We can represent $11$ consecutive integers as $x, x+1, x+2, \ldots, x+10$ . $\newline$ The product of these integers is $x \times (x+1) \times (x+2) \times \ldots \times (x+10)$ . $\newline$ Since the sequence contains $11$ numbers, at least one of them must be even (because every second integer is even). $\newline$ The product of an even number with any other numbers is always even. $\newline$ 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. $\newline$ The correct answer is (B) Only II.