Are you Perfect?

Number Theory Level 2

Which of the following numbers is a perfect square?

\dfrac{17!\times 18!}{2}

\dfrac{16!\times 17!}{2}

\dfrac{14!\times 15!}{2}

\dfrac{15!\times 16!}{2}

2 solutions

Rishabh Jain
Jan 30, 2016

$\large \dfrac{17!\times 18!}{2}=\dfrac{17!\times 17!\times 18}{2}\\ ~~~~~~~~~~~\small{\color{#D61F06}{(Writing~18!=18\times 17!)}}$ $\large =17!\times 17! \times 9$ $\large =(17!\times 3)^2$ $\Large \text{Hence } \color{#69047E}{\boxed{\color{#20A900}{\dfrac{17! \times 18!}{2}}}} \text{ is a perfect square}$

Thành Trần
Feb 5, 2016

17!x18!/2 = (17!)^2x18/2 = (17!x3)^2... It's answer.
15!x16!/2 = (15!)^2x16/2 = (15!)^2x8...
16!x17!/2 = (1x3x...x16)^2x17x4/2 = (1x3x...x16)^2x34...
14!x15!/2 = (1x3x...x14)^2x15x4/2 = (1x3x...x14)^2x30... aren't perfect square.

Are you Perfect?

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