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

## Other useful links

- https://www.cheenta.com/roots-of-equations-prmo-2016-problem-8/
- https://www.youtube.com/watch?v=M_HvBNmPcfU

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