Question

There are 3 consecutive odd numbers and 3 consecutive even numbers. The smallest even number is 9 more than the largest odd number. If the square of the average of odd numbers is 507 less than the square of the average of even numbers, what is the smallest odd number?

• A. 11
• B. 13
• C. 17
• D. 19

Hint:

Use variable expressions for consecutive numbers and apply algebraic identities like \( a^2 - b^2 = (a - b)(a + b) \) to solve efficiently.

Answer 1

The Correct Option is A

Let the three consecutive odd numbers be \( x-2, x, x+2 \).
Then their average is: \[ \frac{(x-2) + x + (x+2)}{3} = \frac{3x}{3} = x \] Let the three even numbers be \( y-2, y, y+2 \). Their average is also \( y \) According to the question: \[ y - 2 = x + 2 + 9 \Rightarrow y = x + 13 \] Also given: \[ y^2 - x^2 = 507 \] Substitute \( y = x + 13 \): \[ (x + 13)^2 - x^2 = 507 \Rightarrow x^2 + 26x + 169 - x^2 = 507 \Rightarrow 26x = 338 \Rightarrow x = 13 \] Then smallest odd number: \[ x - 2 = 13 - 2 = \boxed{11} \]