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Number Theory Level 3

Solve the linear Diophantine equation $2011x + 2017y = 1$ for integer solution such that $x$ is least natural number. Submit your answer as $x + y$ .

The answer is 1.

1 solution

Rajen Kapur
Jan 14, 2016

$\frac{2017}{2011}=1 +\cfrac{1}{335 +\cfrac{1}{6}}$ , ignoring insignificant $\frac{1}{6}$ in comparison to large 335 the given fraction is marginally less than $\frac{336}{335}$ . Equating this to $\frac{x}{-y}$ , we get x = 336, y = -335 and $x + y = 1$

You cannot "ignore" the numbers over there. This is a question of number theory and not of numerical analysis. Instead, by euclidean algorithm, the values of x and y can be easily determined.

Anand Chitrao - 5 years, 5 months ago

The intention behind ignoring insignificant fraction is to emphasize the fact that G.I.F. is used here to arrive at a close ratio with smaller numerator and denominator. In my opinion it is a tool of algebra, neither number theory nor numerical analysis.

Rajen Kapur - 5 years, 5 months ago

Try to solve the equation 2012 x + 2017 y = 1 with your method.

Ricardo Moritz Cavalcanti - 9 months, 3 weeks ago

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The answer is 1.

1 solution

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