If the sum of six consecutive even integers is 138, what is the smallest of the six integers? Check Answer

Let the six consecutive even integers be represented as: - \( x \) (the smallest even integer) - \( x + 2 \) - \( x + 4 \) - \( x + 6 \) - \( x + 8 \) - \( x + 10 \) The sum of these integers can be expressed as: \[ x + (x + 2) + (x + 4) + (x + 6) + (x + 8) + (x + 10) \] Combining like terms, we get: \[ 6x + (2 + 4 + 6 + 8 + 10) = 6x + 30 \] We know that this sum equals 138: \[ 6x + 30 = 138 \] To solve for \( x \), we first subtract 30 from both sides: \[ 6x = 138 - 30 \] \[ 6x = 108 \] Next, we divide both sides by 6: \[ x = \frac{108}{6} = 18 \] Thus, the smallest of the six consecutive even integers is: \[ \boxed{18} \] To verify, we can list the six integers: 18, 20, 22, 24, 26, and 28. Their sum is: \[ 18 + 20 + 22 + 24 + 26 + 28 = 138 \] Since the sum is indeed 138, our solution is confirmed. The smallest integer is \( \boxed{18} \).