Math Is Fun Forum

jozou

Member

Registered: 2011-07-31

Posts: 8

"sum of numbers = product of that numbers" problem

Hi guys,

find all solutions of equation a + b + c = a . b . c, where a ≤ b ≤ c are positive integers.

Well, by trial and error method I found out that 1+2+3 = 1.2.3
Probably there is no other solution, but how can I proove that? I tried case analysis, but unsuccessfully. Any idea? Thanks!

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: "sum of numbers = product of that numbers" problem

Hi jozou;

You are correct that is the only solution but I do not have a proof to that.

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.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,621

Re: "sum of numbers = product of that numbers" problem

hi jozou

The restrictions on a, b and c make it very likely that's the only solution and a proof by exhaustion (trying all cases) should do it.

If a = b = c = 3 then the product is way too big compared with the sum.

Make any one variable bigger than three and the product gets even larger than before.

So you only need to consider cases less than or equal to three.  That about wraps it up for one solution only.

Bob

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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

reconsideryouranswer

Member

Registered: 2011-05-11

Posts: 171

Re: "sum of numbers = product of that numbers" problem

a + b + c = abc

abc - a = b + c

a(bc - 1) = b + c

Last edited by reconsideryouranswer (2011-08-03 02:03:49)

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Bob

Administrator

Registered: 2010-06-20

Posts: 10,621

Re: "sum of numbers = product of that numbers" problem

hi reconsideryouranswer

Where did that line come from?

If c = 3 then b le 1.6666 which would force b to be 1 ??

I think the end of your proof needs a re-think.   

Bob

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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

jozou

Member

Registered: 2011-07-31

Posts: 8

Re: "sum of numbers = product of that numbers" problem

One guy found really nice proof:

Let a ≤ b ≤ c be positive integers, s.t. equation a+b+c = a.b.c holds.

If a > 1, then 4c ≤ a.b.c = a+b+c. So 3c ≤ a+b, what contradicts a ≤ b ≤ c.
So a= 1. Now we have to solve 1+b+c = b.c. Then b.(c-1) = 1+b+c - b = 1 + c, (b-1).(c-1) = 1 + c - (c-1) = 2.
So (b-1).(c-1) = 2. Hence the only solution satistying assumptions is b = 2, c = 3.

jozou

Member

Registered: 2011-07-31

Posts: 8

Re: "sum of numbers = product of that numbers" problem

hi jouzo

If a > 1, then 4c ≤ a.b.c = a+b+c. So 3c ≤ a+b, what contradicts a ≤ b ≤ c.

Don't understand where the 4c bit came from.  Can you expand this please?

Bob

If a > 1, then a ≥ 2. Next b ≥ a, so b ≥ 2. Hence a.b ≥ 4.
So  a+b+c = a.b.c ≥ 4c.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,621

Re: "sum of numbers = product of that numbers" problem

hi

Sorry. I was being stupid.  (too early in the morning for me) I worked it out, so I deleted my post.  Thought you had gone off-line.

Nice proof.  Thanks.

Bob

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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

reconsideryouranswer

Member

Registered: 2011-05-11

Posts: 171

Re: "sum of numbers = product of that numbers" problem

hi reconsideryouranswer

Where did that line come from?

If c = 3 then b le 1.6666 which would force b to be 1 ??

I think the end of your proof needs a re-think.   

Bob

My post has been reedited to show the
correct denominator of (c - 1).

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Bob

Administrator

Registered: 2010-06-20

Posts: 10,621

Re: "sum of numbers = product of that numbers" problem

hi reconsideryouranswer,

Thanks for the edit.  That makes good sense to me now. 

Bob

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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