Suppose $x$, $y$, and $z$ are pairwise coprime integers. Are $yz$ and $x(y+z)$ coprime?

Question

Suppose $x$, $y$, and $z$ are pairwise coprime integers, in the sense that $gcd(x,y) = gcd(x,z) = gcd(y,z) = 1$. Are $yz$ and $x(y+z)$ coprime?

Read the proof of why the 2300-year old Euclidean algorithm gives the gcd of two numbers. From there you can figure out the answer. — P Vanchinathan, Jul 27 '16 at 15:24

score 1 · Accepted Answer · answered Jul 27 '16 at 15:18

1

Yes. Any common prime factor would have to divide either both $y$ and $x$, or $y$ and $y+z$ (therefore also $z$), or $z$ and $x$, or $z$ and $y+z$ (therefore also $y$).

answered Jul 27 '16 at 15:18

Robert Israel

448,999

score 1 · Answer 2 · answered Jul 27 '16 at 15:47

1

By Euclid, $x$ and $\, w := y\!+\!z\,$ are coprime to $\,y,z\,$ so coprime to $\,yz,\,$ so $\,xw\,$ is coprime to $\,yz.$

answered Jul 27 '16 at 15:47

Bill Dubuque

272,048

score 1 · Answer 3 · answered Jul 27 '16 at 16:05

Since $x,y,z$ are mutually coprime, the sets of prime divisors of $x,y,z$ are disjoint.
Assume that $\gcd(yz,x(y+z))\neq 1$: in such a case, $yz$ and $x(y+z)$ must have a common prime divisor, and it must be a prime divisor of $y$ or (exclusive) $z$. In both cases, such a prime divisor cannot divide $x$, hence it has to divide $y+z$, but since $\gcd(y,z)=1$, there is no way.

Suppose $x$, $y$, and $z$ are pairwise coprime integers. Are $yz$ and $x(y+z)$ coprime?

3 Answers3