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**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.

Please help

*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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**phrontister****Real Member**- From: The Land of Tomorrow
- Registered: 2009-07-12
- Posts: 4,592

2**x** = 5**y**

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

Hi;

Thank you for your assistance as well as your patience

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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**Monox D. I-Fly****Member**- Registered: 2015-12-02
- Posts: 857

phrontister wrote:

2

x= 5y

From that we can see that the minimum integer values ofxandyare5and2respectively (2 *5= 5 *2).

The minimum sum ofxandytherefore is5+2= 7.

Ugh... Why have I never thought about this?

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