Greatest common divisor

Number Theory Level pending

Let $a, b, and \ c$ be positive integers such that $\gcd(a,bc)=1$ .

What can I say about $\gcd(a,b) \ and \gcd(a,c)$ = ?

gcd(a,c)=1

only

gcd(a,b)=1

only

gcd(a,b)=1 \ and \ gcd(a,c)

1

none of the above

2 solutions

Hana Wehbi
May 29, 2016

Since $gcd(a,bc)=1$ , then there is no prime that divides both $a \ and\ bc$ .

This means that there is no prime that divides both $a$ and $b$ or both $a$ and $c$ .

So, if there is no prime that divides both $a$ and $b$ or both $a$ and $c$ $\implies$ gcd(a,c)=1 and gcd(a,b)=1

展豪張
May 29, 2016

$\gcd(a,b),\gcd(a,c)|\gcd(a,bc)$
$\therefore\gcd(a,b)=\gcd(a,c)=1$

Since, we don't know what are the values of $a, b, c$ . It is a risk to approach it that way? We need "divides" in our solution, true.

Hana Wehbi - 5 years ago

Sorry I don't understand what you say. Can you elaborate it further?

展豪張 - 5 years ago