Online Course Discussion Forum

MCIII Number Theory 3.22

 
 
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MCIII Number Theory 3.22
by Tina Jin - Friday, 5 January 2024, 3:38 PM
 

Hello,

I need some help on this question, can you give a hint?

Thanks,

Tina Jin

 
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Re: MCIII Number Theory 3.22
by Dr. Kevin Wang - Saturday, 6 January 2024, 12:15 AM
 

Write $x^2$ in two different ways and get $$\lfloor x^2 \rfloor + \{ x^2 \} = \lfloor x\rfloor^2 + 2\lfloor x \rfloor \{x\} + \{ x\}^2.$$ Since $\{ x^2 \} = \{x\}^2$, we get $$\lfloor x\rfloor^2 + 2\lfloor x \rfloor \{x\}.$$  Thus we know that $2\lfloor x \rfloor \{x\}$ must be an integer.  For $1\leq x < 7$, there are only a few possible values for $\lfloor x \rfloor$, and try each of them out.

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Re: MCIII Number Theory 3.22
by Tina Jin - Sunday, 14 January 2024, 11:09 AM
 
I tested cases of floor(x), and in each case there are many solutions with complicated fractions that are hard to square and verify that it satisfies the equation quickly. Is there a way of testing each of the 49 solutions without actually plugging in each and every one of them into the equation?