Floor function

Algebra Level 4

Find the last 7 digits (from left to right) of $\left\lfloor \dfrac { { 10 }^{ 9999 } }{ 10^{ 3333 }+2016 } \right\rfloor$ .

Notation : $\lfloor \cdot \rfloor$ denotes the floor function .

The answer is 4064255.

1 solution

Chew-Seong Cheong
Jul 3, 2016

$\begin{aligned} \left \lfloor \frac {10^{9999}}{10^{3333}+2016} \right \rfloor & = \left \lfloor \frac {10^{9999}+2016^3-2016^3}{10^{3333}+2016} \right \rfloor \\ & = \left \lfloor \frac {(10^{3333}+2016)(10^{6666}-2016\cdot10^{3333}+2016^2)-2016^3}{10^{3333}+2016} \right \rfloor \\ & = \left \lfloor 10^{6666}-2016\cdot10^{3333} + 2016^2 - \frac {2016^3}{10^{3333}+2016} \right \rfloor \\ & = 10^{6666}-2016\cdot10^{3333} + 2016^2 -1 \\ & = \underbrace{999...999}_{3329 \text{ digits}}7984\underbrace{000...000}_{3333 \text{ digits}} + 4064256 - 1 \\ & = \underbrace{999...999}_{3329 \text{ digits}}7984\underbrace{000...000}_{3326 \text{ digits}} \underbrace{\boxed{4064255}}_{7 \text{ digits}} \end{aligned}$

Hi Sir, it would be good if you could put $2016^2=(2000+16)^2=(4000000+64000+256)=4064256$ .

Joshua Chin - 4 years, 11 months ago

Oh, I just used a calculator for that. I think it is not the main purpose of the solution.

Chew-Seong Cheong - 4 years, 11 months ago

Floor function

The answer is 4064255.

1 solution

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