Question

Twice the sum of 5 consecutive odd numbers and thrice the sum of 4 consecutive even numbers is 178. Thrice the sum of 5 consecutive odd numbers and twice the sum of 4 consecutive even numbers is 177. Find the smallest even number.

• A. 2
• B. 4
• C. 6
• D. 8

Answer 1

The correct option is C

6

Let the odd numbers be $x,x+2,x+4,x+6andx+8.$
Their Sum = $5x+20$
Let the even numbers be $y,y+2,y+4andy+6.$
Their Sum = $4y+12$

Given that:
$2(5x+20)+3(4y+12)=178$ and
$3(5x+20)+2(4y+12)=177$

Simplifying these two equations, we get
$5x+6y=51$ ...(1)
$15x+8y=93$ ...(2)
Multiplying (1) by 3,
$15x+18y=153$ ...(3)
Subtracting (2) from (3), we get
$10y=60\Rightarrow y=6$

Substituting $y=6$ in (1), we get
$5x+6\left(6\right)=51\Rightarrow 5x+36=51\Rightarrow 5x=15\Rightarrow x=3.$

Therefore, the odd numbers are 3, 5, 7, 9, 11
and the even numbers are 6, 8, 10, 12.

The smallest of the even numbers is 6.