Number theory Number theory For each equation find solutions

Number theory

Number theory For each equation, find solutions x, y\\in mathbb Zor explain why no solutions exist 91 x + 195 y = 25 91 x + 343 = 28

Solution

The linear equation a, b, c Z

a x + b y = c

has an integer solution in x and y Z

gcd(a, b) | c

so,

(i)

91x + 195y = 25

gcd(91,195) = 13

so to have integral solution,it must divide \"c\" i.e.. 25..

which it doesn\'t..so no integral solution..

(ii)

91x + 343y = 28

gcd(a,b) = gcd(91,343)= 7

which divides \"c\" i.e.. 28 in this case..

finding integral solution..

91x+343y= 28

13x+49y=4

13*11 + 49*(-3) = 4