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.
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 20180903 12:28:09 