Find the remainder and quotient when dividing $10^{k}+1$ by $11$,where $k=1, 2, 3, 4, 5$.

$(1)$ For $k=1$,$10^{1}+1=11$. By Euclid's Division Lemma,$11 = 11 \times 1 + 0$. Thus,the quotient is $1$ and the remainder is $0$.
$(2)$ For $k=2$,$10^{2}+1=101$. By Euclid's Division Lemma,$101 = 11 \times 9 + 2$. Thus,the quotient is $9$ and the remainder is $2$.
$(3)$ For $k=3$,$10^{3}+1=1001$. By Euclid's Division Lemma,$1001 = 11 \times 91 + 0$. Thus,the quotient is $91$ and the remainder is $0$.
$(4)$ For $k=4$,$10^{4}+1=10001$. By Euclid's Division Lemma,$10001 = 11 \times 909 + 2$. Thus,the quotient is $909$ and the remainder is $2$.
$(5)$ For $k=5$,$10^{5}+1=100001$. By Euclid's Division Lemma,$100001 = 11 \times 9091 + 0$. Thus,the quotient is $9091$ and the remainder is $0$.