A number theory problem by Wildan Bagus Wicaksono
x x = 2 0 1 7 − 2 0 1 7 − 2 0 1 7 − ⋱ 2 0 1 6 2 0 1 6 2 0 1 6 = 2 0 1 6 − 2 0 1 6 − 2 0 1 6 − ⋱ 2 0 1 5 2 0 1 5 2 0 1 5
Find the value of x that satisfies both the equations above.
The answer is 1.
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1 solution
Remember: When you combine a system of (nonlinear) equations, you risk introducing extraneous roots. As a contrived example, the system x 2 − x − 1 = 0 , x + 1 = 0 has no solutions, even though we can add both of the equations together to get x 2 = 0 .
So, all that you have shown is "A necessary condition for a solution is x = 1 ". However, you have not shown that this is a sufficient condition.
x = 2 0 1 7 − 2 0 1 7 − 2 0 1 7 − ⋮ 2 0 1 6 2 0 1 6 2 0 1 6 x = 2 0 1 7 − x 2 0 1 6 . . . ( 1
x = 2 0 1 6 − 2 0 1 6 − 2 0 1 6 − ⋮ 2 0 1 5 2 0 1 5 2 0 1 5 x = 2 0 1 6 − x 2 0 1 5 . . . ( 2
( 1 = ( 2
2 0 1 7 − x 2 0 1 6 = 2 0 1 6 − x 2 0 1 5 2 0 1 7 − 2 0 1 6 = x 2 0 1 6 − x 2 0 1 5 1 = x 1 ⇒ x = 1