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## #1 2010-04-05 12:47:45

iamfriendly
Member
Registered: 2009-07-03
Posts: 72

Hi guys, here is the question:

The integer numbers x and y satisfy the equation 2x=5y

Only 1 of the following can be the sum of x and y

2012,         2011,      2010,        2009,           2008

The standard anwser is 2009. The question is why??

Any thought might be a helpful hint to others.

Last edited by iamfriendly (2010-04-05 13:44:11)

EYE AM FRIENDLY, THAT'S O U NEED 2 NO,
psst,
Don't trust strangers, EYE AM FRIENDLY, THAT'S O U NEED 2 NO ..........psst, Don't trust strangers......

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## #2 2010-04-05 14:27:01

phrontister
Real Member
From: The Land of Tomorrow
Registered: 2009-07-12
Posts: 4,600

### Re: Plz Help, please !

2x = 5y
From that we can see that the minimum integer values of x and y are 5 and 2 respectively (2 * 5 = 5 * 2).
The minimum sum of x and y therefore is 5 + 2 = 7.

x and y maintain their integer status if their sum is any multiple of 7...and from the 5 choices only 2009 fits.

Another approach:

2x = 5y
∴ x = 5y/2

Let x + y = z

Plug in x:
5y/2 + y = z
5y + 2y = 2z
7y = 2z

So twice z must be a multiple of 7...only satisfied by 2009.

Same applies if plugging y into the x + y = z equation --> 5 times z must be a multiple of 7 (again only satisfied by 2009).

Last edited by phrontister (2010-04-06 02:14:10)

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## #3 2010-04-06 13:28:02

iamfriendly
Member
Registered: 2009-07-03
Posts: 72

### Re: Plz Help, please !

Hi;

Just last night (in bed, before reading your explanation) I was thinking and I found :

for 2x = 5y ==> 1x=2.5y==> 1y= 0.4x ===> x+y=1.4x

Lowest multiple of 1.4 is 7 then yeah.....

a slight variation of your second method,  basically

Just, like, sharing ideas if you don't mind.
The 1st is really helpful, not that the 2nd one isn't, Thank you
'Night, See you around

Last edited by iamfriendly (2010-04-06 13:31:59)

EYE AM FRIENDLY, THAT'S O U NEED 2 NO,
psst,
Don't trust strangers, EYE AM FRIENDLY, THAT'S O U NEED 2 NO ..........psst, Don't trust strangers......

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## #4 2017-10-04 14:50:14

Monox D. I-Fly
Member
From: Indonesia
Registered: 2015-12-02
Posts: 1,283

### Re: Plz Help, please !

phrontister wrote:

2x = 5y
From that we can see that the minimum integer values of x and y are 5 and 2 respectively (2 * 5 = 5 * 2).
The minimum sum of x and y therefore is 5 + 2 = 7.