1. **State the problem:** We have a set of 7 consecutive odd numbers. 2. Let the smallest odd number be $x$. Then the numbers are: $x, x+2, x+4, x+6, x+8, x+10, x+12$. 3. The largest number is $x + 12$. This number is removed and replaced with 31. 4. The new set becomes: $x, x+2, x+4, x+6, x+8, x+10, 31$. 5. The new mean is given as 13, so: $$\frac{x + (x+2) + (x+4) + (x+6) + (x+8) + (x+10) + 31}{7} = 13$$ 6. Simplify the numerator: $$6x + (2 + 4 + 6 + 8 + 10) + 31 = 6x + 30 + 31 = 6x + 61$$ 7. So: $$\frac{6x + 61}{7} = 13$$ 8. Multiply both sides by 7: $$6x + 61 = 91$$ 9. Subtract 61: $$6x = 30$$ 10. Divide by 6: $$x = 5$$ 11. Find the original largest number: $$x + 12 = 5 + 12 = 17$$ **Final answer:** The original largest odd number was **17**.