Where is 44 or 55?

Number Theory Level 2

Find the remainder when the number below is divided by 11:

$\huge {{{{{{{{45}}^{{{{{{{46}}^{{{{{{47}}^{{{{ {48}}^{{{{49}}^{{{50}}^{51^{.^{.^{54}}}}}}}}}}}}}}}}}}}}}}}}}$

2 solutions

Chew-Seong Cheong
Jun 17, 2016

Let the number be $45^n$ , where $n$ is an integer .

$\begin{aligned} 45^n & \equiv (44+1)^n \text{ (mod 11)} \\ & \equiv 1^n \text{ (mod 11)} \\ & \equiv \boxed{1} \text{ (mod 11)} \end{aligned}$

I think you mean $\pmod{11}$

Sam Bealing - 4 years, 12 months ago

Thanks. Silly me.

Chew-Seong Cheong - 4 years, 12 months ago

Nicely explained (+1).

Abhay Tiwari - 4 years, 12 months ago

Tommy Li
Jun 17, 2016

$45 \equiv 1 \pmod{11}$

${{{{{{{{45}}^{{{{{{{46}}^{{{{{{47}}^{{{{ {48}}^{{{{49}}^{{{50}}^{51^{.^{.^{.^{54}}}}}}}}}}}}}}}}}}}}}}}}}} \equiv 1 \pmod{11}$

Nice solution(+1). I am impressed that you caught the main point of the problem.

Abhay Tiwari - 4 years, 12 months ago

Thanks! It is a fast way to solve the problem.

Tommy Li - 4 years, 12 months ago