Math Is Fun Forum

Denominator

Member

Registered: 2009-11-23

Posts: 220

Products of Two Numbers

Hi guys,

I got a problem at school and I have an answer but I'm not so confident about it :\
Anyways, any help is appreciated and I'ld be very thankful

Problem:
The digits from 1 to 9 are used once each to make two numbers like 12345 and 6789 for example.
Write the numbers that when multiplied together will give you the largest possible answer.

Thanks again!
My answer was 97531 and 8642
It could be right but I don't know
Explanations and reasons would be helpful to

Thanks again

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bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Products of Two Numbers

Hi;

That is not the largest.

Welcome to the forum!

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

gnitsuk

Member

Registered: 2006-02-09

Posts: 121

Re: Products of Two Numbers

To prove this we can use the "rearrangement inequality".

This states that given two sets of positive real numbers:

and

and a given rearrangement of X and Y:

The product (and therefore also the sum - as product is just repeated addition):

is maximized when X' and Y' are both ordered in either ascending or descending order.

To prove this consider the case n = 2:

where

and

then

For the general case for any n let

and

such that

and

suppose that

is a permutation of X with

and

is a permutation of Y such that the product

is maximized.

If there exists a pair

then

(case n = 2)

So interchange of these pairs will increase the product. Well if we interchange all such situations we end up with V and Z ordered in the same way.

So in choosing our numbers we should use a greedy approach:

First choose 9 and 8.

then 7 and 6.

We place the 7 so that the partial products of cross multiplication are maximized so we now have 96 and 87 and so on to give:

9642 * 87531 as the largest product.

Now of course my proof above assumes both sets of numbers have equal cardinality but clearly here we have and odd number of digits and our two subsets are of differing cardinality - but you can fill in the details here, details also which need to be shown to prove that any other splitting of the digits will produce only lesser products e.g. a two digit number times a seven digit number etc etc.

interesting links:

http://en.wikipedia.org/wiki/Inequality … tric_means

and

http://en.wikipedia.org/wiki/Chebyshev%27s_inequality

Denominator

Member

Registered: 2009-11-23

Posts: 220

Re: Products of Two Numbers

Sorry gnitsuk, I didn't understand that :\
I'm only in year 9...
But your answer and bobbym's answer don't agree...
So I checked it and bobbym's answer is right.
Thank's for your explanation gnitsuk and your answer bobbym!

I got another question at school which was to do the same but with the digits 9,8,7,6, and 5 and I was part of the few that got it right
875 x 96
:3

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gnitsuk

Member

Registered: 2006-02-09

Posts: 121

Re: Products of Two Numbers

Hi Denominator,

No worries.

I am confused though,

Bobbym's answer was 87531 * 9642

and my answer was 9642 * 87531

They are the same, they both equal 843973902.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Products of Two Numbers

Hi all;

But your answer and bobbym's answer don't agree...

Not exactly, gnitsuk has the right answer in post #3.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Denominator

Member

Registered: 2009-11-23

Posts: 220

Re: Products of Two Numbers

Oh sorry, my eyes deceive me, I need new glasses :\
Thanks again

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