Large Modulo

Level 2

What is the remainder when $2222^{5555}$ is divided by $7$ ?

1 solution

Jubayer Nirjhor
Dec 23, 2013

$2222^{5555}\equiv 3^{5555} \pmod{7}$

$3^{(7-1)}=3^6\equiv 1\pmod{7} ~~~~~~~~~~~[\text{Fermat's Little Theorem}]$

$3^{5555}=3^{5550}\cdot 3^{5}=\left(3^6\right)^{925}\cdot 3^5\equiv 1^{925}\cdot 3^5=3^5=243\equiv 5\pmod{7}$

$\therefore ~~~ 2222^{5555} \equiv \fbox{5} \pmod{7}$

could anyone tell me what those 3 dots at the end of this proof mean?

mathh mathh - 6 years, 10 months ago

Therefore

Good pic....

ashutosh mahapatra - 6 years, 10 months ago

It means Therefore.

A Former Brilliant Member - 6 years, 10 months ago