# Math Is Fun Forum

Discussion about math, puzzles, games and fun.   Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫  π  -¹ ² ³ °

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## #1 2019-12-01 08:32:09

Bob
Registered: 2010-06-20
Posts: 9,299

### 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

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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## #2 2019-12-01 20:52:00

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

### Re: Questions

bob bundy wrote:

hi Laura,

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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## #3 2019-12-01 21:51:17

Bob
Registered: 2010-06-20
Posts: 9,299

### 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

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

Online