Question

Which of the following Diophantine equations cannot be solved? (Show all work.)

1. 6 x + 51 y = 22
2. 33 x + 14 y = 115
3. 14 x + 35 y = 93

Answer 1

A linear Diophantine equation ax+by=c has a solution (in integrers) if and only if gcd(a, b) divides c.

a) Given equation is 6 x + 51 y = 22

gcd (6, 51)= 3

Since 3 does not divide 22

Therefore this equation cannot be solved (to find integer values of x and y).

b) Given equation is 33x + 14 y =115

gcd (33, 14)= 1

Since 1 divides 22

Therefore this equation can be solved (to find integer values of x and y).

c) Given equation is 14x + 35 y =93

gcd (14, 35)= 7

Since 7 does not divide 93

Therefore this equation cannot be solved (to find integer values of x and y).