chloeholding wrote:
Hi,
Question is as follows:
If n is a positive integer and n^2 is divisible by 96, then the largest positive integer that must divide n is...?
My method was to prime factorize 96 then remove any matching factors and multiply these together to get the answer (24) - not sure about the theory behind it though and if anyone has any rules or theories about factors of x compared to factors of x^2 and x^3 etc. that would be very interesting
Thanks
\(n^2\) is divisible by \(96\) i.e. \(2^5*3\)
What is minimum value for \(n^2\) i.e. which is the smallest number must be multiplied by \(2^5*3\) to make it a perfect square?
We know the powers of all prime numbers must be even in a perfect square.
\(2^5\) if multiplied by \(2\) will result in even power of 2 i.e. \(2^6\)
\(3^1\) if multiplied by \(3\) will result in even power of 3 i.e. \(3^2\)
Thus, we multiplied \(2^5*3\) by \(2*3\). n^2 becomes \(2^6*3^2\), which a square of \(2^3*3\)
\((2^3*3)^2=2^6*3^2\)
Thus, we have our minimum value for n.
\(2^3*3=24\)
The largest integer that divides 24 is 24 itself. Smaller integers would be 2,12,4,6,1. Ans: 24(Supposedly)
Now, to add a little more to the confusion.
It is possible that \(n^2=((2^3*3)^2)^2 \hspace{} OR \hspace{2} ((2^3*3)^{25})^2\)
Making \(n=(2^3*3)^2 \hspace{} OR \hspace{2} (2^3*3)^{25} \hspace{2}\)
Eventually making largest integer that could divide \(n=(2^3*3)^2 OR \hspace{2} (2^3*3)^{25}\)
We don't know the exact value of \(n\). Thus, we need to consider it as minimum possible because the question asks
the largest positive integer that MUST divide n.
Imagine we answered it as \((2^3*3)^{25}\) but in reality, \(n=(2^3*3)\), our answer would be wrong because \((2^3*3)^{25}\) doesn't divide \((2^3*3)\).
At the same time, if we answered it as \((2^3*3)\), it will divide "n", alright. But, even here we can't be too sure whether this is indeed the LARGEST integer that is dividing "n". What if \(n=(2^3*3)^{25}\).
I would have liked the question if it read:
"If n is a positive integer and n^2 is
the smallest integer divisible by 96, then the largest positive integer that must divide n will be?