The Difference is Small
x 3 − y 3 = x y + 6 1
The above equation is satisfied for positive integers x , y .
Find x + y .
The answer is 11.
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3 solutions
Why the > 3 ( x − y ) x y in the second line?
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@Omkar, since x > y , this means that x − y > 0 , which in turn means that ( x − y ) 2 > 0 .
Thus x 2 − 2 x y + y 2 > 0 , and by adding 3 x y to both sides we get x 2 + x y + y 2 > 3 x y . Multiplying both sides by ( x − y ) gives us the desired inequality of ( x − y ) ( x 2 + x y + y 2 ) > 3 ( x − y ) x y
Look, the only way I could solve this is by using the title of the problem itself.Factoring x 3 − y 3 = ( x − y ) ( x 2 + x y + y 2 ) = x y + 6 1 → x 2 + x y + y 2 = x y + 6 1 ; ( B e c a u s e x − y = 1 ) → x 2 + y 2 = 6 1 .By trial and error we get x = 5 , y = 6 → x + y = 5 + 6 = 1 1
It is not fair to make the assumption that x − y = 1 . There could be other solutions with x − y = 1 .
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I can't remember when I wrote this solution.I'll fix it.
done by tail and error method that take values from 1 to 10
because ans is not to much large
From the question, we can easily see that x > y , therefore:
x y + 6 1 = ( x − y ) ( x 2 + x y + y 2 ) > 3 ( x − y ) x y
If x − y > = 2 , then 6 1 > = 5 x y , as a result, 1 2 > = x y > = y ( y + 2 ) and y < = 2 . If we substitute y = 1 , 2 into the equation, we see that it won't work.
If x − y = 1 , when we substitute again, we get: x = 6 and y = 5