Problem arises due overuse of magic - Part (1)

Number Theory Level 4

Ben plays a game: He writes the numbers from 1 to 2004 on a board, selects $n$ numbers, then writes their sum $s \bmod {11}$ on the board, casts a magic chant, so that the previous $n$ selected numbers vanish from the board and starts over again with the numbers, which are left at the board.

selecting $n$ numbers,
writing down the $s \bmod { 11 }$ ,
making the previous $n$ numbers vanish.

At the end there were just two numbers left. One was 1000. What was the other one?

The answer is 4.

1 solution

Dharampal Mahla
Jul 29, 2015

s mod 11 = the remainder when 's' is divided by 11

now the sum of 1 to 2004 = 2009010 (sum of AP)

we know that one no. is 1000 so the remaining sum = 2008010

as we know the remainder can not exceeds 10 it means the missing no. is the

remainder of the remaining sum (2008010) when it is divided by 11

and it's 4

Problem arises due overuse of magic - Part (1)

The answer is 4.

1 solution

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