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Question

The number N =999.....9 consists of exactly 2018 nines. Compute the number N^3. How many nines are there in N^3.

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Solution

Recall that any number consisting solely of k 9's has the form ( (10^k)−1 ) (since it is 1 smaller than the number 10^k),
therefore 99999...with 2018 9's can be represented as N=(10^(2018))-1

N³=((10^2018)-1)³
=(10^2018)^3 - 3(10^2018)² + 3(10^2018) - 1

=(10^6054)-3(10^4036)+3(10^2018)-1
=(10^6054-1) -3(10^4036)+3(10^2018)

((10^6054)-1) will contain 6054 zeros.

But adding 3(10^2018) will replace 9 by 3 at 2018th 9's position.
So number of 9 is reduced to 6053.

Now Subtracting 3(10^4036) will replace 9 by 7 at 4036th 9's position.
​​​​​​so number of 9 is reduced to 6052.

Therefore N³ will contain 6052 9's.

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