Show that $forall_{n,m,kin mathbb{N}}(k|nm$ and gcd$(n,k)=1 Rightarrow k|m)$
up vote
1
down vote
favorite
Please help me
Definitons
gcd := greatest common divisor
Let $k,nin mathbb{Z}$ One says that $k$ divides $n$, in symbols $k|n$ if there exists a $l in mathbb{Z}$ s.t. $n=kl$
For two natural numbers $m,n in mathbb{N}$ the greatest common divisior is defined (symblolic gcd$(k,n)$), the biggest natural number $k$ with $k|m$ and $k|n$
algebra-precalculus prime-numbers
asked Nov 21 at 11:15
RM777
818
add a comment |
up vote
1
down vote
favorite
Please help me
Definitons
gcd := greatest common divisor
Let $k,nin mathbb{Z}$ One says that $k$ divides $n$, in symbols $k|n$ if there exists a $l in mathbb{Z}$ s.t. $n=kl$
For two natural numbers $m,n in mathbb{N}$ the greatest common divisior is defined (symblolic gcd$(k,n)$), the biggest natural number $k$ with $k|m$ and $k|n$
algebra-precalculus prime-numbers
asked Nov 21 at 11:15
RM777
818
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Please help me
Definitons
gcd := greatest common divisor
Let $k,nin mathbb{Z}$ One says that $k$ divides $n$, in symbols $k|n$ if there exists a $l in mathbb{Z}$ s.t. $n=kl$
For two natural numbers $m,n in mathbb{N}$ the greatest common divisior is defined (symblolic gcd$(k,n)$), the biggest natural number $k$ with $k|m$ and $k|n$
algebra-precalculus prime-numbers
asked Nov 21 at 11:15
RM777
818
Please help me
Definitons
gcd := greatest common divisor
Let $k,nin mathbb{Z}$ One says that $k$ divides $n$, in symbols $k|n$ if there exists a $l in mathbb{Z}$ s.t. $n=kl$
For two natural numbers $m,n in mathbb{N}$ the greatest common divisior is defined (symblolic gcd$(k,n)$), the biggest natural number $k$ with $k|m$ and $k|n$
algebra-precalculus prime-numbers
algebra-precalculus prime-numbers
asked Nov 21 at 11:15
RM777
818
asked Nov 21 at 11:15
RM777
818
asked Nov 21 at 11:15
RM777
818
asked Nov 21 at 11:15
RM777
818
asked Nov 21 at 11:15
RM777
818
818
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
up vote
2
down vote
accepted
If $m = 0$ the implication is obvious
If $m ne 0$
Then we calculate $gcd(nm,km) = m*gcd(n,k) = m$ becquse $gcd(n,k) = 1$
We know thqt $k|nm$ and $k|km$ thus $k|gcd(nm,km)$
And $gcd(mn,mk) = m$
$$implies k|m$$
answered Nov 21 at 12:39
TheD0ubleT
1784
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
2
down vote
accepted
If $m = 0$ the implication is obvious
If $m ne 0$
Then we calculate $gcd(nm,km) = m*gcd(n,k) = m$ becquse $gcd(n,k) = 1$
We know thqt $k|nm$ and $k|km$ thus $k|gcd(nm,km)$
And $gcd(mn,mk) = m$
$$implies k|m$$
answered Nov 21 at 12:39
TheD0ubleT
1784
add a comment |
up vote
2
down vote
accepted
If $m = 0$ the implication is obvious
If $m ne 0$
Then we calculate $gcd(nm,km) = m*gcd(n,k) = m$ becquse $gcd(n,k) = 1$
We know thqt $k|nm$ and $k|km$ thus $k|gcd(nm,km)$
And $gcd(mn,mk) = m$
$$implies k|m$$
answered Nov 21 at 12:39
TheD0ubleT
1784
add a comment |
up vote
2
down vote
accepted
up vote
2
down vote
accepted
If $m = 0$ the implication is obvious
If $m ne 0$
Then we calculate $gcd(nm,km) = m*gcd(n,k) = m$ becquse $gcd(n,k) = 1$
We know thqt $k|nm$ and $k|km$ thus $k|gcd(nm,km)$
And $gcd(mn,mk) = m$
$$implies k|m$$
answered Nov 21 at 12:39
TheD0ubleT
1784
If $m = 0$ the implication is obvious
If $m ne 0$
Then we calculate $gcd(nm,km) = m*gcd(n,k) = m$ becquse $gcd(n,k) = 1$
We know thqt $k|nm$ and $k|km$ thus $k|gcd(nm,km)$
And $gcd(mn,mk) = m$
$$implies k|m$$
answered Nov 21 at 12:39
TheD0ubleT
1784
answered Nov 21 at 12:39
TheD0ubleT
1784
answered Nov 21 at 12:39
TheD0ubleT
1784
answered Nov 21 at 12:39
TheD0ubleT
1784
1784
add a comment |
add a comment |
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3007576%2fshow-that-forall-n-m-k-in-mathbbnknm-and-gcdn-k-1-rightarrow-km%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
