100 Day Summer Challenge

100 problems in 100 days. #100problems

Day 100

100 of 100: One Hundred and One

The first few powers of 101 all begin with 1, as highlighted by red colors below:

1011=1011012=102011013=10303011014=104060401 \begin{aligned} 101^1 &= {\color{#D61F06}{1}}01 \\ 101^2 &= {\color{#D61F06}{1}}0201 \\ 101^3 &= {\color{#D61F06}{1}}030301 \\ 101^4 &= {\color{#D61F06}{1}}04060401 \\ &\vdots& \end{aligned}

Is this always the case for all positive integer powers of 101?

Bonus: If it is the case, why? If not, what is the smallest power of 101 that doesn't start with a 1?

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