Question

The $5^{99}$ when divided by $13$

Answer 1

The number $5$ follows a fixed pattern when divided by $13$ the remainder is

$5^1=5$

$5^2=12$

$5^3=8$

$5^4=1$

$5^5=5$

$5^6=12$

$5^7=8$

$5^8=1$

$5^9=5$

$5^{10}=12$

......

$5^{97}=5$

$5^{98}=12$

$5^{99}=8$

To find the remainder when $5^{99}$ is divided by $13$ we follow the above pattern after every four intervals starting from $5^1$ the remainder $5$ is repeated hence we see $5^1=5,5^5=5,5^9=5,5^{13}=5,5^{17}=5,5^{21}=5,5^{25}=5,...5^{97}=5$

We see for $(1-10)$, we have $1,5,9$ as powers of $5$ where remainder is $5$ when divided by $13$

For $(10-20)$ we have $13,17$ as powers of $5$ where remainder $=5$ when divided by $13$

Following this pattern for other intervals,

$(20-30),21,25,29$

$(31-40),33,37$

$(41-50),41,45,49$ and so on.

We get for $(91-100),93,97$

Hence $5^{99}$ when divided by $13$ will get remainder $8$