Step 1: Let the smallest of the 8 consecutive numbers be \( x \). The numbers are \( x, x+1, x+2, \ldots, x+7 \).
Step 2: Sum \( = 8x + (0+1+2+\cdots+7) = 8x + 28 \).
Step 3: Set the sum equal to 108: \( 8x + 28 = 108 \Rightarrow 8x = 80 \Rightarrow x = 10 \).
Step 4: The largest number \( = x + 7 = 17 \).
Answer: 17.