I.S.I. and C.M.I. Entrance

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

Try this beautiful Hundred Integers problem on Number system from TOMATO useful for ISI B.Stat Entrance. You may use sequential hints to solve the problem.

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



Check the Answer

But try the problem first…

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

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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