Question

find the solution of the linear Diophantinc equation 172x + 20y = 1000​

Answer 1

Answer:

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Step-by-step explanation:

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Answer 2

Answer:

the particular solution of the given equation is $172(500) + 20(-4250) = 1000$

Step-by-step explanation:

Given a linear Diophantine equation 172x + 20y = 1000

Diophantine equation is a polynomial in the form of $ax + by = c$, with two or more unknowns, and the only solutions of interest are the integer ones.

Let us find the greatest common divisor(GCD) of 172 and 46 using division method.

$20\big)~172 ~\big(8\\~~~~~~\underline{~160~}\\~~~~~~~~12\big)~20 ~\big(1\\~~~~~~~~~~~~~\underline{~12~}\\~~~~~~~~~~~~~~~8\big)~12 ~\big(1\\~~~~~~~~~~~~~~~~~~\underline{~~8~}\\~~~~~~~~~~~~~~~~4\big)~8 ~\big(2\\~~~~~~~~~~~~~~~~~~\underline{~~8~}\\~~~~~~~~~~~~~~~~~~~~0$

Last non-zero remainder the GCD of the numbers, so here the GCD of 172 and 46 is 4.

Hence $d=GCD(172, 46)= 4.$

Writing the above division method in terms of remainder, we get 3 equations.

So, using the formula,

Remainder = Dividend - Divisor × Quotient

$12=172-20. 8$               ...(1)

$8=20-12. 1$                  ...(2)

$4=12-8. 1$                    ...(3)

Now to solve for x and y, let us write the extended Euclidean algorithm.

For that, let us take equation (3) and substitute equation (2) in it:

$4=12-[20-12. 1]. 1$

$\implies 4=12-[20.1-12. 1]$

$\implies 4=12. 2-[20.1]$

Now substituting equation (1) in the above equation,

$4=[172-20. 8]. 2-[20.1]$

$\implies 4=[172. 2-20. 16]-[20.1]$

$\implies 4=[172. 2-20. 17]$

Comparing this with the given equation 172x + 20y = 1000

we get $x=2$ and $y =-17$

And for the constant, c in the equation, the condition is $c = dp\implies p=\frac{c}{d}$

Therefore, $p=\dfrac{1000}{4} =250$

Hence, the given equation is equivalent to 172(2) + 20(-17) = 4(250)

For particular solutions of the given equation, find $x^{' }=x\times p$ and $y^{' }=y\times p$ are

$\implies x^{' }=2\times 250=500$ and $y^{' }=(-17)\times 500=-4250$

Therefore, the particular solution of the given equation is $172(500) + 20(-4250) = 1000$