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Ones vs twos (Posted on 2018-09-03) Difficulty: 1 of 5
Each of the integers 1 to 10,000,000 is repeatedly replaced by its digital sum until we obtain a list of 10^7 decimal digits (a.k.a. digital roots).

Will this list have more ones or twos?

Justify your answer.

No Solution Yet Submitted by Ady TZIDON    
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analytical solution | Comment 1 of 4
summing a numbers digits preserves its value mod 9.  Thus the original number and its final single digit value will have the same value mod 9.  


looking at the numbers from 1 to 10,000,000 we can break them up into blocks of 9.


10000000/9=1111111 + 1/9


Thus there will be 1111111 full blocks of 9 numbers and a final block with a single number (namely 10000000).


10000000=1 mod 9 thus it will have a final value of 1


Thus 1 will appear 1111112 times while the rest of the values will appear 1111111 times.


Thus 1 will appear more often.

  Posted by Daniel on 2018-09-03 12:28:09
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