That's a lot of variables... what do I do?

Number Theory Level 4

Let $\psi$ be defined as a function such that for any positive integer $n$ , $\psi (n)$ finds the smallest possible integral value for $m \geq 2$ such that $n^{m} \pmod{10}=n \pmod{10}$ . Let $P$ be the probability that if a random nonnegative integer $n$ is selected that $\psi (n)=5$ . If $P$ can be written as $\frac{a}{b}$ where $a$ and $b$ are coprime, positive integers, find $a+b$ .

The answer is 7.

2 solutions

Daniel Liu
Apr 7, 2014

We need only consider when $n\equiv 0\to 9\pmod{10}$ , so let's check the cases for $n=0\to 9$ .

Note that for $n=2,3,7,8$ , we have that $n^5\equiv n\pmod{10}$ :

$\boxed{2},4,8,16,3\boxed{2}$

$\boxed{3},9,27,81,24\boxed{3}$

$\boxed{7},49,343,2401,1680\boxed{7}$

$\boxed{8},64,512,4096, 3276\boxed{8}$

For all the others, $m<5$ .

Therefore, our probability is $\dfrac{4}{10}=\dfrac{2}{5}$ and our desired answer is $2+5=\boxed{7}$ .

Perfect! :D

Finn Hulse - 7 years, 2 months ago

It seems to me the question is not well stated. In all cases m=1 or m=0 are the lowest integer values that work.

Since these are trivial, one has to assume the author meant m>1.

Steven Perkins - 7 years, 2 months ago

Thanks. I have updated the problem accordingly.

In future, if you spot any errors with a problem, you can “report” it by selecting the “dot dot dot” menu in the lower right corner. You will get a more timely response that way.

Calvin Lin Staff - 6 years, 5 months ago

Ramiel To-ong
Jun 5, 2015

total percentage of P = 40%

That's a lot of variables... what do I do?

The answer is 7.

2 solutions

0 pending reports