Try this beautiful hundred integers problem based on Remainder useful for ISI B.Stat Entrance.

Hundred Integers | ISI B.Stat Entrance | Problem-82


Let \(x_1,x_2,……,x_100\) be hundred integers such that the sum of any five of them is 20. Then..

  • the largest \(x_i\) equals 5
  • the smallest \(x_i\) equals to 3
  • \(x_{17} = x_{83}\)
  • none of the foregoing statements is true

Key Concepts


Number theory

Divisor

integer

Check the Answer


But try the problem first…

Answer:\(x_{17} = x_{83}\)

Source
Suggested Reading

TOMATO, Problem 82

Challenges and Thrills in Pre College Mathematics

Try with Hints


First hint

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 ……….

Second Hint

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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