5. In a $100 \times 100$ table, ten thousand numbers are written arbitrarily. Let $a_{1}$ denote the sum of the numbers in the first column, $a_{2}$ the sum of the numbers in the second column, and so on, with $a_{100}$ the sum of the numbers in the hundredth column. Furthermore, let $b_{1}$ denote the sum of the numbers in the first row, $b_{2}$ the sum of the numbers in the second row, and so on, with $b_{100}$ the sum of the numbers in the hundredth row. Calculate the value of the expression:

$$
\left(a_{1}-b_{1}\right)+\left(a_{2}-b_{2}\right)+\ldots+\left(a_{100}-b_{100}\right)
$$
Solution. The given expression can be written in the form

$$
\left(a_{1}+a_{2}+\ldots+a_{100}\right)-\left(b_{1}+b_{2}+\ldots+b_{100}\right)
$$

and it is equal to zero, because the sum in the first parentheses is the sum of all 10000 numbers (the sum of the columns), and the sum in the second parentheses is also equal to the sum of all 10000 numbers (the sum of the rows).

## Eighth Section