Is it possible to arrange the integers from 1 to 240 in a table with 15 rows and 16 columns so that the sum of the numbers in each of the columns would be the same?
To determine if it is possible to arrange the integers from 1 to 240 in a table with 15 rows and 16 columns such that the sum of the numbers in each column is the same, we need to check if the total sum of the numbers from 1 to 240 is divisible by the number of columns.
The sum of the integers from 1 to 240 can be calculated using the formula for the sum of an arithmetic series:
Sum = (n/2)(first term + last term),
where n is the number of terms.
In this case, the first term is 1, the last term is 240, and the number of terms, n, can be calculated by multiplying the number of rows (15) by the number of columns (16):
n = 15 * 16 = 240.
Plugging these values into the formula, we get:
Sum = (240/2)(1 + 240) = 120 * 241 = 28920.
Now, we need to check if 28920 is divisible by the number of columns, which is 16. We can do this by dividing 28920 by 16 and checking if the remainder is 0.
Dividing 28920 by 16, we get:
28920 ÷ 16 = 1807 remainder 8.
Since there is a remainder of 8, it means that the sum of the numbers from 1 to 240 cannot be evenly distributed among 16 columns. Therefore, it is not possible to arrange the integers in the desired way.