Prove that the equation $18x+42y=22$ has no integer solution?

Question

Can you help me to prove? Should I prove it by contradiction and suppose that $x$, y are solutions ?

What integer divides the LHS but not the RHS, and what can you conclude from that? — Bill Dubuque, Dec 21 '16 at 20:24
This is a very classical diophantine equation ; I am surprised that answers don't refer to sites like this one ; ) where one can find explanations/proofs — Jean Marie, Dec 21 '16 at 20:35
It would be nice if, in the future, you let us know what you know about the problem, or what you have tried. At the very least, you could start by reducing to $9x+21y=11$, for example. It would also be helpful to know what sort of class this is for... — The Count, Dec 21 '16 at 20:39
@JeanMarie Surely because a proof of this direction of the equivalence is quite trivial (unlike the opposite), so there is no need for such. — Bill Dubuque, Dec 21 '16 at 20:41
I reduced to 9x+21y=11 then because left side can be divided by 3 and right cannot I assumed there was no solution but when it comes to maths I never trust my self :c thanks for advice, I will do better next time ^ — Сона Хачунц, Dec 21 '16 at 20:43

score 8 · Answer 1 · answered Dec 21 '16 at 20:25

8

$ax+by=c$ has solution in the integers if and only if $\mathrm{gcd}(a,b)$ divides $c$. Hence no solution in your example.

(In particular, $18x+42y$ is multiple of $3$, while $22$ not.)

answered Dec 21 '16 at 20:25

Paolo Leonetti

score 4 · Answer 2 · answered Dec 21 '16 at 20:27

4

Notice that for integers $x$ and $y$, the left-hand side is always a multiply of $3$; but $22$...

$$18x+42y=22 \iff \color{blue}{3} \left( 6x+14y \right) = 22$$

answered Dec 21 '16 at 20:27

StackTD

score 1 · Answer 3 · answered Dec 22 '16 at 12:36

1

Observe that LHS is a multiple of 3 but RHS is not a multiple of 3 so there exist rational solutions but not integral solutions

answered Dec 22 '16 at 12:36

3 Answers3