Self-counting machine

The self-counting machine works exactly like a calculator. The innkeeper wanted to add several three-digit natural numbers on his own. On the first attempt, he got the result in 2224. To check, he added these numbers again, and he got 2198. Therefore, he added these numbers again, and now he got the sum of 2204. He always entered the fifth added number incorrectly because he needed to press its digits harder on each attempt. In the self-calculation, he always entered a two-digit number instead of a three-digit number. He did not make any other errors in the census; this time, the self-count also worked flawlessly. What is the correct sum of inn numbers?

Final Answer:

s =  2324

Step-by-step explanation:

$$\begin{gathered} 2224=s-(a\cdot 100+b\cdot 10+c)+b\cdot 10+c \\ 2198=s-(a\cdot 100+b\cdot 10+c)+a\cdot 10+b \\ 2204=s-(a\cdot 100+b\cdot 10+c)+a\cdot 10+c \\ 0<a<10 \\ 0\le b<10;0\le c<10 \\ s=2324 \\ x=139 \\ a=1 \\ b=3 \\ c=9 \\ {c}_1=2324-139+39=2224 \\ {c}_2=2324-139+13=2198 \\ {c}_3=2324-139+19=2204 \end{gathered}$$

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