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**Bob****Administrator**- Registered: 2010-06-20
- Posts: 9,276

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

Children are not defined by school ...........The Fonz

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

Sometimes I deliberately make mistakes, just to test you! …………….Bob

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**Monox D. I-Fly****Member**- From: Indonesia
- Registered: 2015-12-02
- Posts: 2,000

bob bundy wrote:

hi Laura,

Who is this Laura you're talking about? Is this thread originally a reply to another thread?

Actually I never watch Star Wars and not interested in it anyway, but I choose a Yoda card as my avatar in honor of our great friend bobbym who has passed away.

May his adventurous soul rest in peace at heaven.

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**Bob****Administrator**- Registered: 2010-06-20
- Posts: 9,276

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

Children are not defined by school ...........The Fonz

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

Sometimes I deliberately make mistakes, just to test you! …………….Bob

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