Briefly justify the following statement there are no integer

Briefly justify the following statement: there are no integers, a, b Z, such that x + y = 116 and gcd(x, y) = 12.

Solution

Since gcd(x,y) =12 ,both x and y are divisible by 12

Let x=12a

Y= 12b

Thus, 12a+12b=116

12(a+b)=116

a+b!= Integer value

Thus x and y cannot be integers.

Hence proved