Concatenated Pythagoras Triplet

Number Theory Level 4

Let $N=\overline{345345345..........}$ be a $300$ digit number. What is the remainder when N is divided by $999$ ?

1 solution

Chew-Seong Cheong
Oct 7, 2014

$N$ can be expressed as:

$N = 345(1 + 10^3 + 10^6+...+10^{297}$

It is noted that $10^{3i} = (1+999)^i \Rightarrow 10^{3i} \equiv 1 \pmod {999}$

$\Rightarrow N \equiv 345(100) \pmod {999} \equiv [34(1+999) +500] \pmod {999} \equiv \boxed {534} \pmod {999}$

awesome and fantastico!!

Adarsh Kumar - 6 years, 8 months ago

$\color{#D61F06}{S}$ $\color{#3D99F6}{UUUUUUUUUUU}$ $\color{#20A900}{P}$ $\color{#624F41}{E}$ $\color{#333333}{R}$ $\color{#D61F06}{B}$

jaiveer shekhawat - 6 years, 7 months ago

Did the same!

Kartik Sharma - 6 years, 6 months ago