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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

But try the problem first…

Answer:$x_{17} = x_{83}$

Source

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 $x_{17} =x_{83}$