How to find all integer pairs $(x,y)$ that satisfy the equation $21x+47y=1$ without using modular arithmetic?

Question

Find all integer pairs $(x,y)$ that satisfy the following equation: $$ 21x+47y=1. $$

I know the solution with modular arithmetic: \begin{align} 0x + 5y &= 1 \pmod{21} \\ 5y &= -20 \pmod{21}\\ y = -4 \pmod{21} &\Longleftrightarrow y = -4 + 21c \\ 21x + 47(-4 + 21c) &= 1\\ 21x &= 189 - 21 * 47c \\ x &= 9 - 47c. \\ \end{align}

Therefore $$(x, y) = (9 - 47c, -4 + 21c)$$

Is there a way to do this without modular arithmetic? Thanks in advance!

Please use LaTeX formatting so your question is readable! Why do you want to do this without modular arithmetic? — The Count, Dec 21 '16 at 20:52
it is the same kind of question you have asked some minutes ago ! — Jean Marie, Dec 21 '16 at 20:52
Please Stop. You have asked two same questions today. http://math.stackexchange.com/questions/2067728/prove-that-the-equation-18x42y-22-has-no-integer-solution. — Behrouz Maleki, Dec 21 '16 at 20:52
I feel like I will never be able to use stackexchange correctly :o I'm sorry — Сона Хачунц, Dec 21 '16 at 21:02

score 1 · Answer 1 · answered Dec 21 '16 at 20:58

1

Hint: $\operatorname{gcd}(21,47)=1$ so there is a solution, by Bézout's identity. Suppose $(x,y)$ is such a solution. Then note \begin{align} 21x+47y=1&\iff21x+21\cdot47+47y-21\cdot47=1\\ &\iff 21(x+47)+47(y-21)=1 \end{align}

answered Dec 21 '16 at 20:58

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How to find all integer pairs $(x,y)$ that satisfy the equation $21x+47y=1$ without using modular arithmetic?

1 Answers1