here is the problem:(adsbygoogle = window.adsbygoogle || []).push({});

1. what is the remainder if 11..........1 is divided by 1001? (the dividend has 1000 1's)

this is what i did (please tell me if there is any other methods of doing it):

[tex]11............1[/tex]

[tex]=10^{999} + 10^{998} + \cdots + 10^{0}[/tex]

[tex]=\frac{10^{999} - 1}{9}[/tex]

i noticed that

[tex]1000 = 10^{3} \equiv -1 \left(\bmod 1001\right)[/tex]

[tex]10^{6} \equiv 1 \left(\bmod 1001\right)[/tex]

[tex]10^{996} \equiv 1 \left(\bmod 1001\right)[/tex]

[tex]10^{996}.10^{4} \equiv 10^{4} \equiv 991 \left(\bmod 1001\right)[/tex]

[tex]10^{1000} - 1 \equiv 990 \left(\bmod 1001\right)[/tex]

[tex]\frac{10^{999} - 1}{9} \equiv 110 \left(\bmod 1001\right)[/tex]

is there any problem in my method? is there any other easier way of doing it?

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# A problem (and my solution)

**Physics Forums | Science Articles, Homework Help, Discussion**