Confusing step in proof of divisibility by $7$.
$begingroup$
I am reading the proof of the divisibility rule for $7$ here (Aops page),but I can't see how $k-2n_0 equiv 2n_0 +6k $ is derived.
I've tried to derive it by this: $$2n_0 +6k equiv 2n_o +7k-k equiv 2n_0-k, $$
but how can I get from this to $2n_0+6k equiv k-2n_0$?
elementary-number-theory proof-explanation
edited Dec 22 '18 at 18:57
Shaun
9,854113684
asked Jan 23 '16 at 18:20
Mr. YMr. Y
1,374723
$endgroup$
add a comment |
$begingroup$
I am reading the proof of the divisibility rule for $7$ here (Aops page),but I can't see how $k-2n_0 equiv 2n_0 +6k $ is derived.
I've tried to derive it by this: $$2n_0 +6k equiv 2n_o +7k-k equiv 2n_0-k, $$
but how can I get from this to $2n_0+6k equiv k-2n_0$?
elementary-number-theory proof-explanation
edited Dec 22 '18 at 18:57
Shaun
9,854113684
asked Jan 23 '16 at 18:20
Mr. YMr. Y
1,374723
$endgroup$
$begingroup$
$$7mid 10k+n_0iff 7mid 2(10k+n_0) =20k+2n_0$$ $$iff 7mid (20k+2n_0)-7(3k)=-k+2n_0$$
$endgroup$
– user236182
Jan 23 '16 at 18:27
$begingroup$
thanks,but why in the comments ?You don't give me the opportunity to accept your answer and show my gratitude :).
$endgroup$
– Mr. Y
Jan 23 '16 at 18:42
$begingroup$
@user236182 Please type up your hints as an answer in order to close the question. Until then (or whenever someone else decides to answer in greater detail), this community wiki answer referencing you will suffice :)
$endgroup$
– Shaun
Dec 22 '18 at 18:25
add a comment |
$begingroup$
I am reading the proof of the divisibility rule for $7$ here (Aops page),but I can't see how $k-2n_0 equiv 2n_0 +6k $ is derived.
I've tried to derive it by this: $$2n_0 +6k equiv 2n_o +7k-k equiv 2n_0-k, $$
but how can I get from this to $2n_0+6k equiv k-2n_0$?
elementary-number-theory proof-explanation
edited Dec 22 '18 at 18:57
Shaun
9,854113684
asked Jan 23 '16 at 18:20
Mr. YMr. Y
1,374723
$endgroup$
I am reading the proof of the divisibility rule for $7$ here (Aops page),but I can't see how $k-2n_0 equiv 2n_0 +6k $ is derived.
I've tried to derive it by this: $$2n_0 +6k equiv 2n_o +7k-k equiv 2n_0-k, $$
but how can I get from this to $2n_0+6k equiv k-2n_0$?
elementary-number-theory proof-explanation
elementary-number-theory proof-explanation
edited Dec 22 '18 at 18:57
Shaun
9,854113684
asked Jan 23 '16 at 18:20
Mr. YMr. Y
1,374723
edited Dec 22 '18 at 18:57
Shaun
9,854113684
asked Jan 23 '16 at 18:20
Mr. YMr. Y
1,374723
edited Dec 22 '18 at 18:57
Shaun
9,854113684
edited Dec 22 '18 at 18:57
Shaun
9,854113684
edited Dec 22 '18 at 18:57
Shaun
9,854113684
9,854113684
asked Jan 23 '16 at 18:20
Mr. YMr. Y
1,374723
asked Jan 23 '16 at 18:20
Mr. YMr. Y
1,374723
asked Jan 23 '16 at 18:20
Mr. YMr. Y
1,374723
1,374723
$begingroup$
$$7mid 10k+n_0iff 7mid 2(10k+n_0) =20k+2n_0$$ $$iff 7mid (20k+2n_0)-7(3k)=-k+2n_0$$
$endgroup$
– user236182
Jan 23 '16 at 18:27
$begingroup$
thanks,but why in the comments ?You don't give me the opportunity to accept your answer and show my gratitude :).
$endgroup$
– Mr. Y
Jan 23 '16 at 18:42
$begingroup$
@user236182 Please type up your hints as an answer in order to close the question. Until then (or whenever someone else decides to answer in greater detail), this community wiki answer referencing you will suffice :)
$endgroup$
– Shaun
Dec 22 '18 at 18:25
add a comment |
$begingroup$
$$7mid 10k+n_0iff 7mid 2(10k+n_0) =20k+2n_0$$ $$iff 7mid (20k+2n_0)-7(3k)=-k+2n_0$$
$endgroup$
– user236182
Jan 23 '16 at 18:27
$begingroup$
thanks,but why in the comments ?You don't give me the opportunity to accept your answer and show my gratitude :).
$endgroup$
– Mr. Y
Jan 23 '16 at 18:42
$begingroup$
@user236182 Please type up your hints as an answer in order to close the question. Until then (or whenever someone else decides to answer in greater detail), this community wiki answer referencing you will suffice :)
$endgroup$
– Shaun
Dec 22 '18 at 18:25
$begingroup$
$$7mid 10k+n_0iff 7mid 2(10k+n_0) =20k+2n_0$$ $$iff 7mid (20k+2n_0)-7(3k)=-k+2n_0$$
$endgroup$
– user236182
Jan 23 '16 at 18:27
$begingroup$
$$7mid 10k+n_0iff 7mid 2(10k+n_0) =20k+2n_0$$ $$iff 7mid (20k+2n_0)-7(3k)=-k+2n_0$$
$endgroup$
– user236182
Jan 23 '16 at 18:27
$begingroup$
thanks,but why in the comments ?You don't give me the opportunity to accept your answer and show my gratitude :).
$endgroup$
– Mr. Y
Jan 23 '16 at 18:42
$begingroup$
thanks,but why in the comments ?You don't give me the opportunity to accept your answer and show my gratitude :).
$endgroup$
– Mr. Y
Jan 23 '16 at 18:42
$begingroup$
@user236182 Please type up your hints as an answer in order to close the question. Until then (or whenever someone else decides to answer in greater detail), this community wiki answer referencing you will suffice :)
$endgroup$
– Shaun
Dec 22 '18 at 18:25
$begingroup$
@user236182 Please type up your hints as an answer in order to close the question. Until then (or whenever someone else decides to answer in greater detail), this community wiki answer referencing you will suffice :)
$endgroup$
– Shaun
Dec 22 '18 at 18:25
add a comment |
1 Answer
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$begingroup$
This community wiki answer is to point out that this comment, posted by @user236182 (who is invited to post an answer), answers the question.
Summarising it:
$$begin{align}
7mid 10k+n_0 &iff 7mid 2(10k+n_0) =20k+2n_0 \
&iff 7mid (20k+2n_0)-7(3k)=-k+2n_0.
end{align}$$
$endgroup$
add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
This community wiki answer is to point out that this comment, posted by @user236182 (who is invited to post an answer), answers the question.
Summarising it:
$$begin{align}
7mid 10k+n_0 &iff 7mid 2(10k+n_0) =20k+2n_0 \
&iff 7mid (20k+2n_0)-7(3k)=-k+2n_0.
end{align}$$
$endgroup$
add a comment |
$begingroup$
This community wiki answer is to point out that this comment, posted by @user236182 (who is invited to post an answer), answers the question.
Summarising it:
$$begin{align}
7mid 10k+n_0 &iff 7mid 2(10k+n_0) =20k+2n_0 \
&iff 7mid (20k+2n_0)-7(3k)=-k+2n_0.
end{align}$$
$endgroup$
add a comment |
$begingroup$
This community wiki answer is to point out that this comment, posted by @user236182 (who is invited to post an answer), answers the question.
Summarising it:
$$begin{align}
7mid 10k+n_0 &iff 7mid 2(10k+n_0) =20k+2n_0 \
&iff 7mid (20k+2n_0)-7(3k)=-k+2n_0.
end{align}$$
$endgroup$
This community wiki answer is to point out that this comment, posted by @user236182 (who is invited to post an answer), answers the question.
Summarising it:
$$begin{align}
7mid 10k+n_0 &iff 7mid 2(10k+n_0) =20k+2n_0 \
&iff 7mid (20k+2n_0)-7(3k)=-k+2n_0.
end{align}$$
answered Dec 22 '18 at 18:24
community wiki
Shaun
add a comment |
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$begingroup$
$$7mid 10k+n_0iff 7mid 2(10k+n_0) =20k+2n_0$$ $$iff 7mid (20k+2n_0)-7(3k)=-k+2n_0$$
$endgroup$
– user236182
Jan 23 '16 at 18:27
$begingroup$
thanks,but why in the comments ?You don't give me the opportunity to accept your answer and show my gratitude :).
$endgroup$
– Mr. Y
Jan 23 '16 at 18:42
$begingroup$
@user236182 Please type up your hints as an answer in order to close the question. Until then (or whenever someone else decides to answer in greater detail), this community wiki answer referencing you will suffice :)
$endgroup$
– Shaun
Dec 22 '18 at 18:25