<![CDATA[Math Is Fun Forum / Questions]]> 2019-12-02T09:51:17Z FluxBB https://www.mathisfunforum.com/viewtopic.php?id=25370 <![CDATA[Re: Questions]]> Sorry for the confusion.  Someone emailed me with two questions and it was easier to post my answers here so I could get math formatting.

The questions were:

Q1.  What is the smallest number that has 30 positive factors?

I've given an answer but I'm not certain I've fully proved it yet.  It helped that the email had a suggested answer but asked how do we prove it's the lowest?

Q2.  (x^2 - y^2) = P a prime >= 2.  I'm unsure if this required the lowest values  or just any values which is why I explored several.

Bob

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https://www.mathisfunforum.com/profile.php?id=67694 2019-12-02T09:51:17Z https://www.mathisfunforum.com/viewtopic.php?pid=411568#p411568
<![CDATA[Re: Questions]]> bob bundy wrote:

hi Laura,

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https://www.mathisfunforum.com/profile.php?id=212031 2019-12-02T08:52:00Z https://www.mathisfunforum.com/viewtopic.php?pid=411564#p411564
<![CDATA[Questions]]> hi Laura,

Q1. 1x2x3x4x5x6 makes 720 and that has 30 factors so I like your Dad's answer.  Is it the lowest?

Well if you substitute any larger number in place of one of {1,2,3,4,5,6} then you'd get a larger product and maybe not even 30 factors.

If you leave out one and repeat one of {1,2,3,4,5,6} you'll get less factors.

If you don't use all six then it definitely won't have more factors.

So the 'proof' is you've found a contender and then shown there's nothing better.

Q2.  Do you recognise the expression 'difference of two squares' ?

That looks like two factors then, so not a prime.

But every number can be written as 1 x the number so the only way you can get a prime is if (x-y) = 1

So how about x = 3, y = 2 ….

So that's the lowest solution. EDIT: Actually x = 2 y = 1 is lower.

But others are also possible such as x = 7 and y = 6

But not all x-y = 1.  eg. x=41 y = 40   gives 81 which is not prime.

Hope that helps Bob

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https://www.mathisfunforum.com/profile.php?id=67694 2019-12-01T20:32:09Z https://www.mathisfunforum.com/viewtopic.php?pid=411554#p411554