Question

solve the linear diophantine equation 172x+20y=1000
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Answer 1

Answer:

100x23y = - 19 if and only if 23y = 100x + 19, if and only if 100x + 19 is divisible by 23. Using modular arithmetic, you have

100x + 19 = 0 (mod 23) 100x19 (mod

8x 4

(mod 23)

2x = 1 (mod 23)

X = 12 (mod 23).

s

Answer 2

The solution for the given linear diophantine equation 172x+20y=1000 is x = 500 and y = - 4250

Given,

a = 172

b = 20

c = 1000

To Find,

The values of x and y

Solution,

We have been given the linear diophantine equation 172x+20y=1000

Let gcd(a,b) = d

The given diophantine equation will have a solution only if d divides c

We apply Euclid's algorithm to find d = gcd(172,20)

$172 = 20 \times 8 \hspace{0.1cm} + 12\\20 = 12 \times 1 \hspace{0.1cm} + 8\\12 = 8 \times 1 \hspace{0.1cm} + 4\\8 = 4 \times 2 \hspace{0.1cm} + 0$

that is, d = 4

4 divides 1000

Therefore, the given linear diophantine equation has a solution

Now again, by retracing Euclid's algorithm, we can find the solution

$4 = 12 - 8\\4 = 12 - (20 - 12)\\4 = 12 - 20 - 12\\4 = 12 \times 2 - 20\\4 = (172 - 20 \times 8) \times 2 - 20\\4 = 172 \times 2 - 20 \times 16 - 20\\4 = 172 \times 2 + 20 \times {-17}$

Multiply both sides by 250

$1000 = 172 \times 500 + 20 \times {-4250}$

Hence, x = 500 and y = - 4250

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