Solve-the-equation-71a-3b-2021-

Question Number 127532 by MathSh last updated on 30/Dec/20

S o l v e t h e e q u a t i o n :

71 a + 3 b = 2021

Commented by mr W last updated on 30/Dec/20

s e e Q 44819

Answered by Ar Brandon last updated on 30/Dec/20

2021 = 28 (71) + 33

= 71 (28) + 3 (11)

(a, b) = (28, 11) is a solution

Commented by MathSh last updated on 30/Dec/20

T h a n k s s i r

Answered by floor(10²Eta[1]) last updated on 30/Dec/20

General solution to a Diophantine equation .

ax + by = c

let d = \gcd (x, y) \Rightarrow d ∣ x, d ∣ y ∴ d ∣ ax + by = c

ax + by has a solution if and only if d ∣ c .

now suppose (x_{0}, y_{0}) is a particular solution,

then,

{ax}_{0} + {by}_{0} = ax + by = c

a (x_{0} - x) = b (y - y_{0})

since d = \gcd (a, b) \Rightarrow a = a^{'} d, b = b^{'} d

where \gcd (a^{'}, b^{'}) = 1

{da}^{'} (x_{0} - x) = {db}^{'} (y - y_{0})

a^{'} (x_{0} - x) = b^{'} (y - y_{0})

\Rightarrow (I) : a^{'} ∣ b^{'} (y - y_{0}) \Rightarrow a^{'} ∣ (y - y_{0})

[because \gcd (a^{'}, b^{'}) = 1 \Rightarrow a^{'} ∤ b^{'}]

\Rightarrow y - y_{0} = a^{'} k \Rightarrow y = y_{0} + a^{'} k

\Rightarrow (II) : b^{'} ∣ a^{'} (x_{0} - x) \Rightarrow b^{'} ∣ (x_{0} - x)

\Rightarrow x_{0} - x = b^{'} k \Rightarrow x = x_{0} - b^{'} k

so the general solution to a diophantine

equation of the form ax + by = c is

(x, y) = (x_{0} - \frac{bk}{d}, y_{0} + \frac{ak}{d}), k \in Z

where (x_{0}, y_{0}) is a particular solution

now that you know this try to solve

your question .

Commented by MathSh last updated on 31/Dec/20

T h a n k s s i r

Facebook Tweet Pin