### Consider the $800$-digit integer$234523452345 ..... 2345.$ The first $m$ digits and the last $n$ digits of the above integer are crossed out so that the sum of the remaining digits is $2345$. FInd the value of $m + n$.

$130$

Step by Step Explanation:
1. Observe that the given number has $2345$ repeated $200$ times.
$2 + 3 + 4 + 5 = 14$
The sum of digits of the given number $= 14 \times 200 = 2800$
2. After crossing out the first $m$ digits and the last $n$ digits, the sum is $2345$.
$\implies$ the sum of first $m$ and last $n$ digits is $2800 - 2345 = 455$
3. Observe that $455 = 32 \times 14 + 7$. Thus we have to cross out $32$ blocks of $4$ digits $2345$ either from the front or the back, a $2$ from the front that remains and a $5$ from the back that remains. Thus, $m + n = 32 \times 4 + 2 = 130$
4. Hence, the value of $m + n$ is $130$.