Question

For each of the following linear Diophantine equations either finde
solutions or show that there are no integral solutions.
(i) 2x + 5y = 11
17x + 13y = 100
21x+ 14y = 147
(iv) 60x + 18y = 97

​

Answer 1

Answer:

Step-by-step explanation:

Bro According to Your Questions Your Answer is

(a) First (2,5) = 1 divides 11. so by Theorem 3.23, there are infinitely

many solutions. To find these solutions note that by Euclidean Algorithm

2(-2) +5 = 1 and so 2(-22) +5(11) = 11. Thus, I = -22 and y = 11 is a

particular solution. All solutions are given by

I = ro + (b/d)n=-22+5n, and y = 90 - (a/dl)n = 11 - 2n

(b) First (17.13) = 1 divides 100. so by Theorem 3.23, there are infinitely

many solutions. To find these solutions note that by Euclidean Algorithm

17(-3)+13(4) = 1 and so 17(–300) + 13(400) = 100. Thus, I = – 300 and

y = 400 is a particular solution. All solutions are given by

I = 10 + (b/d)n = -300+13n, and y = y0 - (a/d)n = 400 - 17n

(c) First (21.14) = 7 divides 147, so by Theorem 3.23, there are infinitely

many solutions. To find these solutions note that by Euclidean Algorithm

21+14(-1) = 7 and so 21(21)+14(-21) = 147. Thus, I = 21 and y = -21

is a particular solution. All solutions are given by

I = 10 + (b/d)n=21+ 2n, and y=yu - (a/d)n= -21 - 3n

2.

(d) Because (60,18) = 6 does not divide 97, it follows that there are no

integral solutions of the Diophantine Equation.

(e) First 1402=2.701 and 1969 = 11. 179, and so (1402, 1969) = 1 divides

1, so by Theorem 3.23, there are infinitely many solutions. To find these

solutions note that by Euclidean Algorithm 1402(889) + 1969(-633) = 1.

Thus, I = 889 and y = -633 is a particular solution. All solutions are given

by

<= 10 + (b/d)n = 889+ 1969n , and y=yo - (a/d)n= -633 - 1402n

RESULT

(a) 2 = -22 +5n, and y= 11 – 2n.

(b) 1 = -300+ 13n, and y = 400 - 17n.

(c)2 = 21 + 2n , and y=-21 - 3n.

(d) No integral solutions.

(e) x = 889+ 1969n, and y=-633 - 1402n.