Question

What is the remainder when 2^(495) is divided by 11? No calculator allowed. Please show technique

Answer 1

hmmm...
maybe we can try to use fermat's little theorem

Answer 2

\[495=11(45) \\ 2^{495}=2^{11 \cdot 45 }\]
then later we can use that 45=4(11)+1

Answer 3

recall fermat's little theorem is
\[a^p \equiv a (\mod p)\]
p is prime

Answer 4

I literally have no idea what you are doing lol

Answer 5

have you ever heard of fermat's little theorem

Answer 6

\[2^{495} \equiv (2^{11})^{45}\equiv 2^{45} \text{ by fermat's little theorem } \\ 2^{45} \equiv2^{11(4)+1}\equiv(2^{11})^42^{1} \equiv (2)^4(2)\equiv =2^5 \equiv 32 \]
do you what 32 mod 11 is?

Answer 7

by the way i used fermat's little theorem twice

Answer 8

\[2^{495} \equiv (2^{11})^{45}\equiv 2^{45} \text{ by fermat's little theorem } \\ 2^{45} \equiv2^{11(4)+1}\equiv(2^{11})^42^{1} \equiv (2)^4(2) \text{ by fermat's little theorem again }\\ (2)^4(2)=2^5 \equiv 32 \equiv ?\]
the last step I'm asking you to do what 32 mod 11?

Answer 9

@liliy have you left me already?