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 Post subject: Prove that there exists a 3-digit natural numberPosted: Sun, 27 Sep 2009 00:03:42 UTC
 S.O.S. Oldtimer

Joined: Fri, 18 Jul 2008 01:21:42 UTC
Posts: 147
Location: California, USA
Hello,

Hope someone can help me.

Problem: Prove that there exists a 3-digit natural number less than 400 w/distinct digits, such that the sum of the digits is 17 and the product of the digits is 108.

My setup: I have 2 equations:

(1) xyz = 108
(2) x +y +z = 17.

I did some substitution and the last step I got was x^2z +xz^2 -17xz +108. This is where I got stuck. I can't factor by grouping. Could someone give me a hint?

Thank you.

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Susan

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 Post subject: Re: Prove that there exists a 3-digit natural numberPosted: Sun, 27 Sep 2009 00:16:56 UTC
 Member of the 'S.O.S. Math' Hall of Fame

Joined: Thu, 1 Jan 2004 06:47:21 UTC
Posts: 2735
Location: Norman, Oklahoma
Susan123456 wrote:
Hello,

Hope someone can help me.

Problem: Prove that there exists a 3-digit natural number less than 400 w/distinct digits, such that the sum of the digits is 17 and the product of the digits is 108.

My setup: I have 2 equations:

(1) xyz = 108
(2) x +y +z = 17.

I did some substitution and the last step I got was x^2z +xz^2 -17xz +108. This is where I got stuck. I can't factor by grouping. Could someone give me a hint?

Thank you.

Have you learned Lagrange Multipliers?

Dave

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 Post subject: Posted: Sun, 27 Sep 2009 00:58:54 UTC
 S.O.S. Oldtimer

Joined: Fri, 18 Jul 2008 01:21:42 UTC
Posts: 147
Location: California, USA
Yes, I have but it's been a while. I wonder if I can use Algebra for this proof. Thank you.

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Susan

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 Post subject: Posted: Sun, 27 Sep 2009 01:12:21 UTC
 Member of the 'S.O.S. Math' Hall of Fame

Joined: Thu, 1 Jan 2004 06:47:21 UTC
Posts: 2735
Location: Norman, Oklahoma
Susan123456 wrote:
Yes, I have but it's been a while. I wonder if I can use Algebra for this proof. Thank you.

I don't think that'd be the best way to go.

Dave

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 Post subject: Re: Prove that there exists a 3-digit natural numberPosted: Sun, 27 Sep 2009 02:54:18 UTC
 Moderator

Joined: Mon, 29 Dec 2008 17:49:32 UTC
Posts: 6004
Location: 127.0.0.1, ::1 (avatar courtesy of UDN)
Susan123456 wrote:
Hello,

Hope someone can help me.

Problem: Prove that there exists a 3-digit natural number less than 400 w/distinct digits, such that the sum of the digits is 17 and the product of the digits is 108.

My setup: I have 2 equations:

(1) xyz = 108
(2) x +y +z = 17.

I did some substitution and the last step I got was x^2z +xz^2 -17xz +108. This is where I got stuck. I can't factor by grouping. Could someone give me a hint?

Thank you.

Since , x,y,z can only be 1,2,3,4,6,8 or 9. Because you want your 3-digit nmber to be less than 400, and there are no 2 single-digit numbers that give a product >81, the hundred digit (presumably x?) must be 2 or 3. Now examine each case.

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 Post subject: Re: Prove that there exists a 3-digit natural numberPosted: Sun, 27 Sep 2009 07:18:06 UTC
 S.O.S. Oldtimer

Joined: Fri, 27 Jul 2007 10:17:26 UTC
Posts: 278
Location: Chandler, AZ, USA
Lagrange Multipliers? That's opening a peanut with a sledgehammer!

Spoiler:
Since 100x+10y+z<400, x<4 so that x=1, 2 or 3

x=1 means yz=108 (impossible for one-digit numbers).

x=3 means yz=36 and y+z=14 (no such combination of integers exists).

x=2 means yz=54 and y+z=15, thus (y,z) = (6,9) or (9,6)

so 269 and 296 are the numbers we seek.

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 Post subject: Posted: Sun, 27 Sep 2009 08:35:34 UTC
 Moderator

Joined: Mon, 29 Dec 2008 17:49:32 UTC
Posts: 6004
Location: 127.0.0.1, ::1 (avatar courtesy of UDN)
Actually, it is easier to establish one of the digits is 6.

From the factorisation of 108, you know the three digits must contain at least one even number. Also, since the sum of all three digits is odd, we know the three digits must be 2 even and one odd. Now look at how the three 3s can distribute, and you are forced to conclude one of the digit must be 6.

So you are reduced to solving for two numbers whose sum is 11, and whose product is 18. So solving gives you the two numbers.

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 Post subject: Posted: Sun, 27 Sep 2009 17:44:57 UTC
 S.O.S. Oldtimer

Joined: Fri, 18 Jul 2008 01:21:42 UTC
Posts: 147
Location: California, USA

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Susan

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