Online Course Discussion Forum

8 Week AIME Sprint Camp

 
 
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8 Week AIME Sprint Camp
by Ethan Sun - Sunday, 3 December 2023, 5:22 PM
 

For the week one homework I don't know how to do #3...

I have absolutely no way to approach it right now, we didn't really cover problems like this in the class


Thx :)

 
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Re: 8 Week AIME Sprint Camp
by John Lensmire - Monday, 4 December 2023, 1:39 PM
 

Let me start with a hint that hopefully helps :)

In the problem we want $\lfloor \sqrt{n} \rfloor$ to be a factor of $n$. Let's try to take advantage of the fact that floors are always an integer.

If $k = \lfloor \sqrt{n} \rfloor$, then note that $k \leq \sqrt{n} < k+1$ (think about how we round down for the floor). Hence, $k^2 \leq n < k^2 + 2k + 1$. From here, try to consider how $k$ can be a factor of $n$. The advantage we have now is that we know both $k$ and $n$ are integers.

Hope this helps! If necessary, try to look at a few values of $k$ and try to find a pattern.

Picture of Ethan Sun
Re: 8 Week AIME Sprint Camp
by Ethan Sun - Monday, 4 December 2023, 5:34 PM
 

Thanks! This was very helpful and I was able to solve the question!!!