Question

Find all integer solutions to 14x + 77y = 69

Answer 1

What is the GCF or GCD of 14 and 77?

Answer 2

7

Answer 3

So we can rewrite the left side to go from

14x+77y

to

7(2x+11y)


If x and y are integers, then 2x and 11y are integers. Consequently, 2x+11y is also an integer.

So we have something like this:

7*(some integer) = 69


But this implies that 7 is a factor of 69....which is NOT true

Answer 4

So it's simply not possible to have x and y be integers and satisfy the equation 14x+77y = 69

Answer 5

nice :D
I never thought of that @jim_thompson5910 ^_^

Answer 6

yes, nice reasoning indeed

Answer 7

Today I learned a good question:)

Answer 8

So in general, the rule is this

if Ax+By = C is true, where A, B, C, x and y are integers, then the GCD of A and B must divide (or must be a factor of) the right side C

Answer 9

If the GCD is not a factor, then there are no integer solutions.

Answer 10

Which theorem is this?
Plz tell @jim_thompson5910 :)

Answer 11

hmm I want to say some theorem with "gauss" in the name of it....but the full name escapes me atm

Answer 12

oh i see
btw thanx for ur nice solution:)

Answer 13

yw