Can Anyone Help Me?

I am struck with the questions which include Diophantine equations. I have a lot of them(questions).The question asks :

Find the number of pairs (x,y) of positive integral solutions for the following equation.

$2x+3y = 763$

Please provide Solution.

#Algebra #DiophantineEquations

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`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
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`\sum_{i=1}^3`	$\sum_{i=1}^3$
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Comments

Hint: $2x + 3y = 2x + 3y - 6n + 6n = 2(x - 3n) + 3(y + 2n) = 763$ . And find the smallest and largest integer solution of $x$ that satisfy the original equation.

Pi Han Goh - 5 years, 8 months ago

What have you tried? Where did you get stuck?

Check out Linear Diophantine Equations.

Calvin Lin Staff - 5 years, 8 months ago

Sir, isn't there a much time saving way ? Because finding the first random solutions takes a lot time even if you go by the Division Algorithm.

Vishal Yadav - 5 years, 8 months ago

Not particularly.

For small numbers, you can guess. E.g. $y = 1, x = \frac{760}{2}$ works. You can also just try $x = 1, 2, 3, \ldots$ and hopes it works out (which it eventually will).

The Extended Euclidean Algorithm for an algorithm that guarantees finding the values within a reasonable amount of time. I think it is on the order of $O ( \log n )$ , instead of testing all values which has the order of $O(n)$ .

Calvin Lin Staff - 5 years, 8 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$