       Re: Find two numbers a,b such us: a+b=5432 & LCM[a,b]=223020

• To: mathgroup at smc.vnet.net
• Subject: [mg122158] Re: Find two numbers a,b such us: a+b=5432 & LCM[a,b]=223020
• From: János Löbb <janos at lobb.com>
• Date: Mon, 17 Oct 2011 08:09:17 -0400 (EDT)
• Delivered-to: l-mathgroup@mail-archive0.wolfram.com
• References: <201110162045.QAA19688@smc.vnet.net>

```Well, a newbie approach would be something like this:

lcm=223020
a+b=5432

let n such as n*a=lcm and let m such that m*b=lcm  where n and m are Integers.

Then a=lcm/n and b=lcm/m

Then lcm/n + lcm/m = 5432 or

223020*(1/n +1/m) = 5432

Punch this into Matematica and use Reduce:

In:= Reduce[{223020(1/n + 1/m)==5432},{n,m}, Integers]
Out= (n==59&&m==135)||(n==135&&m==59)

Writing these back to a and b, we get a= {3780,1652} and b={1652,3780}

J=E1nos

On Oct 16, 2011, at 4:45 PM, dimitris wrote:

> This is taken from the recent book (2010): " Mathematica: A Problem-
> Centered Approach" by R. Hazrat.
>
> "The sum of two positive integers is 5432 and their least common
> multiple is
> 223020. Find the numbers."
>
> A solution:
>
> Do[If[LCM[i, 5432 - i] == 223020, Print[i, "  ", 5432 - i]], {i, 1,
> 2718}]
> 1652  3780
>
> I wonder if we can solve the system of equations:
>
> a+b==5432&&LCM[a,b]==223020
>
> using codes that contain built in functions like Reduce.
>
> I guess this is not a trivial one because the so much powerful Reduce
> itself fails
>
> In:= Reduce[{a + b == 5432, LCM[a, b] == 223020}, {a, b}, Integers]
>
> During evaluation of In:= Reduce::nsmet:This system cannot be
> solved with the methods available to Reduce. >>
>
> Out= Reduce[{a + b == 5432, LCM[a, b] == 223020}, {a, b}, Integers]
>
> Any ideas?
>
> Dimitris
>

```

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