The Basic Principle

Question

In any n+1 integers there will be a pair which differs by a multiple of n.

I have tried to create a pigeon hole with numbers a0,a1,a2,...,an but i could not get a solution.

Answer 1

Hint. When you take numbers modulo $n$ (that is, find the remainder after dividing by $n$), there are only $n$ possible results. Two numbers that share the same result differ by...

Answer 2

Any number can be written in term of n as Number = n * Quotient + remaineder Here number is a1, a2 and so on , Quotient be anything depends on number But Remainder always lies between 0 to n-1(total count n). So , Diff of two number=diff of Quotient * n + diff of remainder

Diff of two number is not equal to multiple of n if and only if diff of remainder is not equal to 0 or remainder should be unique. But it impossible to generate unique remainder other than 0 to n-1.