+0

# Find the remainder when r^13 + 1 is divided by r-1.

0
728
2
+644

Find the remainder when r^13 + 1 is divided by r-1.

Dec 4, 2017

#1
+98125
+1

Performing synthetic division, we have

1  [   1    0    0    0    0    0   0    0   0   0   0  0   0    1 ]

1    1    1    1    1   1    1   1   1   1  1   1    1

____________________________________

1    1    1    1    1     1   1   1   1    1   1  1   1   2

The remainder is 2

Dec 4, 2017
edited by CPhill  Dec 4, 2017
#2
+21848
+2

Find the remainder when r^13 + 1 is divided by r-1.

Geometric sequence:

$$\begin{array}{|rcll|} \hline \displaystyle 1+r+r^2+r^3+r^4+\ldots +r^{12} &=& \displaystyle\frac{r^{13}-1}{r-1} \\\\ \displaystyle 1+r+r^2+r^3+r^4+\ldots +r^{12} &=& \displaystyle\frac{r^{13}}{r-1}- \frac{1}{r-1} \quad & | \quad + \frac{2}{r-1} \\\\ \displaystyle 1+r+r^2+r^3+r^4+\ldots +r^{12} +\frac{2}{r-1} &=& \displaystyle\frac{r^{13}}{r-1}- \frac{1}{r-1} + \frac{2}{r-1} \\\\ \displaystyle 1+r+r^2+r^3+r^4+\ldots +r^{12} +\frac{2}{r-1} &=& \displaystyle\frac{r^{13}}{r-1}+ \frac{1}{r-1} \\\\ \displaystyle 1+r+r^2+r^3+r^4+\ldots +r^{12} +\frac{\color{red}2}{r-1} &=& \displaystyle\frac{r^{13}+1}{r-1} \\ \hline \end{array}$$

The remainder is 2.

Dec 4, 2017