Question

What is the remainder when ${13}^{17}+{17}^{13}$ is divided by 10

Answer 1

The remainder when dividing on integer by $10$ always equal the units digit Ignore all but the units digit and rephrase the question

What is the units digits of $3^{17}+7^{13}$,

1) The pattern for the units digits of $3$ in $[3,9,7,1]$ every fourth term is the some. The ${17}^{th}$ power is $1$ past the end in $1,3^{17}$ must end in $3$.

2) The pattern for the units digit of $7$ is $[7,9,3,1]$

every fourth term is the some. The ${13}^{th}$ power is $1$ part the end of the repeat since $7^{12}$ ends in $1,7^{13}$ must end in $7$

$\therefore$ Remainder when ${13}^{17}+{17}^{13}$ is $3+7=10$