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19 was removed from the first 33 counting numbers.
N-1 must be a multiple of 16, since we are dealing with natural numbers and the average of N-1 numbers is a rational number which reduces to have 16 as the denominator.
Also, the average for the first n natural numbers is just n/2. 16²=256, so 271/16 is just a little more than 16. By inspection, we can guess that there were 33 numbers, with an average of a little more than 16, and after 1 is erased, the average of the remaining n-1 numbers is also relatively close to 16.
For 32 numbers to have an average of 271/16, their sum must be 271×2=542. The total of the first 33 numbers is 561, a difference of 19, so 19 must have been the number that was removed.
Charlie posted a solution based on the algebraic formulas involved, here.
Also, DJ posted a general method for solving this problem given any value for the average; that can be found here. |
