Question

1+6¹+6².....+6¹⁰⁰ find unit digit

Answer 1

The unit digit of all numbers written in the form $6^n$ where $n\in\mathbb{N}$ is always 6.

$\longrightarrow6^n\equiv6\pmod{10}$

We're asked to find unit digit of the series $1+6^1+6^2+6^3+\,\dots\,+6^{100}.$

Each term in the series except 1 has unit digit 6.

$\longrightarrow6^1\equiv6\pmod{10}$

$\longrightarrow6^2\equiv6\pmod{10}$

$\longrightarrow6^3\equiv6\pmod{10}$

$\vdots$

$\longrightarrow6^{100}\equiv6\pmod{10}$

There are 100 powers of 6 in this series.

Hence the unit digit of the sum will be given by,

$\longrightarrow1+6^1+6^2+6^3+\,\dots\,+6^{100}\equiv1+\underbrace{6+6+\,\dots\,+6}_{100\ terms}\pmod{10}$

$\longrightarrow1+6^1+6^2+6^3+\,\dots\,+6^{100}\equiv1+6\times100\pmod{10}$

$\longrightarrow1+6^1+6^2+6^3+\,\dots\,+6^{100}\equiv601\pmod{10}$

Since $601\equiv1\pmod{10},$

$\longrightarrow\underline{\underline{1+6^1+6^2+6^3+\,\dots\,+6^{100}\equiv1\pmod{10}}}$

Hence 1 is the unit digit.