Try this beautiful hundred integers problem based on Remainder useful for ISI B.Stat Entrance.
Let \(x_1,x_2,......,x_100\) be hundred integers such that the sum of any five of them is 20. Then..
Number theory
Divisor
integer
Answer:\(x_{17} = x_{83}\)
TOMATO, Problem 82
Challenges and Thrills in Pre College Mathematics
Let us take the numbers be \(x_i , x_j , x_k ,x_l , x_m \)
Now \(x_i + x_j + x_k + x_l + x_m = 20\) and again \(x_i + x_j + x_k + x_l + x_n = 20\)
Can you now finish the problem ..........
From the above relation there are three case arise that....
1)\(x_m = x_n\)
2)All the integers are equal.
3)\(x_{17} =x_{83}\)
So the correct answer is \(x_{17} =x_{83}\)
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