Adding Zeros

Number Theory Level 1

101 × 11 = 1111 1001 × 11 = 11011 10001 × 11 = 110011 1000001 × 11 = ? \large{ \begin{array} {c c l c c } 101&\times&11&=&1111 \\ 1001&\times&11&=&11011 \\ 10001&\times&11&=&110011 \\ 1000001&\times&11 &=& ?\end{array} }


The answer is 11000011.

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Rabeya Hossain by Rabeya Hossain , 20, Bangladesh

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3 solutions

Ashish Menon
May 22, 2016

1 01 × 11 = 11 0 11 = 1111 1 001 × 11 = 11 0 0 11 = 11011 1 0001 × 11 = 11 0 0 0 11 = 110011 ∴ 1 000001 × 11 = 11 00 0 00 11 = 11000011 \begin{aligned} 1\color{#D61F06}{0}1 × \color{#EC7300}{11} & = \color{#20A900}{11}{\cancel{\color{#D61F06}{0}}}\color{#20A900}{11} & = \color{#3D99F6}{1111}\\ \\ 1\color{#D61F06}{00}1 × \color{#EC7300}{11} & = \color{#20A900}{11}{\cancel{\color{#D61F06}{0}}}\color{#D61F06}{0}\color{#20A900}{11} & = \color{#3D99F6}{11011}\\ \\ 1\color{#D61F06}{000}1 × \color{#EC7300}{11} & = \color{#20A900}{11}\color{#D61F06}{0}{\cancel{\color{#D61F06}{0}}}\color{#D61F06}{0}\color{#20A900}{11} & = \color{#3D99F6}{110011}\\ \\ \therefore 1\color{#D61F06}{00000}1 × \color{#EC7300}{11} & = \color{#20A900}{11}\color{#D61F06}{00}{\cancel{\color{#D61F06}{0}}}\color{#D61F06}{00}\color{#20A900}{11} & = \boxed{\color{#3D99F6}{11000011}} \end{aligned}

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Justin Malme
Jun 4, 2016

101 X 11 = 1111
1001 X 11 = 11011
10001 X 11 = 110011
Thus,
100001 X 11 = 1100011
1000001 X 11 = 11000011



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Rabeya Hossain
May 22, 2016

It can be noticed that the product is always consists of 11 in the first and last.and the number of 0 between the first and last 11 is one less from the first factor's number of 0.so the answer must be 11000011

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