Anik's remainders

Number Theory Level 3

Anik likes division and is always eager to find remainders.

Once Anik divided $a^x$ by $(a-1)$ to get positive remainder $r_1$ ; and by $(a+1)$ to get positive remainder $r_2$ .

If Anik assumed $a$ as a positive integer greater than 2 and $x$ as an positive odd number greater than 1, find $r_1+r_2$ .

ax

a^x

a-1

a+1

a

ax+1

1 solution

Nihar Mahajan
Jun 9, 2015

We crack this problem using simple congruence:

$a \equiv 1 \pmod{a-1} \\ \Rightarrow a^x \equiv 1 \pmod{a-1} \\ \Rightarrow \boxed{r_1=1}$

Now we will use the fact that $x$ is odd now:

$a \equiv -1 \pmod{a+1} \\ \Rightarrow a^x \equiv -1 \equiv a \pmod{a+1} \\ \Rightarrow \boxed{r_2=a}$

Thus , $\huge\boxed{r_1+r_2=a+1}$ .