Triangular numbers and gcdProving sum of a set is $0 pmod n$ if $n$ is odd, or $fracn2 pmod n$ if $n$ is even?Is greatest common divisor of two numbers really their smallest linear combination?GCD, LCM RelationshipProve a set of nonnegative integers with greatest common divisor 1 and closed under addition has all but finite many nonnegative integers.all pairs of a and b in an equation containing gcdTriangular Numbers Modulo $k$ - Hit All Values?Understanding the Existence and Uniqueness of the GCDGCD and LCM with logical symbolsThe greatest common divisor of two positive integers less than 100 is equal to 3. Their least common multiple is twelve times one of the integers.Suppose that for all integers $x$, $x|a$ and $x|b$ if and only if $x|c$. Then $c = gcd(a,b)$Which is the gcd of 2 numbers which are multiplied and the result is 600000?
Can compressed videos be decoded back to their uncompresed original format?
Simple macro for new # symbol
Processor speed limited at 0.4 Ghz
How to travel to Japan while expressing milk?
ssTTsSTtRrriinInnnnNNNIiinngg
Do Iron Man suits sport waste management systems?
How does having to sign to support someone for elections fit with having a secret ballot?
Why would the Red Woman birth a shadow if she worshipped the Lord of the Light?
How do I gain back my faith in my PhD degree?
Migrate from Force.com IDE to VScode
Explaination of a justification: additive functors preserve limits
Is there an online compendium of Rav Moshe teshuvos in English that exists?
How could indestructible materials be used in power generation?
Why is the sentence "Das ist eine Nase" correct?
What is the opposite of "eschatology"?
Am I breaking OOP practice with this architecture?
Detention in 1997
Should I tell management that I intend to leave due to bad software development practices?
How do conventional missiles fly?
How to stretch the corners of this image so that it looks like a perfect rectangle?
Can I set a Ready action to trigger when literally anything happens?
How to properly check if the given string is empty in a POSIX shell script?
Does the Idaho Potato Commission associate potato skins with healthy eating?
meaning of 腰を落としている
Triangular numbers and gcd
Proving sum of a set is $0 pmod n$ if $n$ is odd, or $fracn2 pmod n$ if $n$ is even?Is greatest common divisor of two numbers really their smallest linear combination?GCD, LCM RelationshipProve a set of nonnegative integers with greatest common divisor 1 and closed under addition has all but finite many nonnegative integers.all pairs of a and b in an equation containing gcdTriangular Numbers Modulo $k$ - Hit All Values?Understanding the Existence and Uniqueness of the GCDGCD and LCM with logical symbolsThe greatest common divisor of two positive integers less than 100 is equal to 3. Their least common multiple is twelve times one of the integers.Suppose that for all integers $x$, $x|a$ and $x|b$ if and only if $x|c$. Then $c = gcd(a,b)$Which is the gcd of 2 numbers which are multiplied and the result is 600000?
$begingroup$
For all positive integers $n$, the $n$th triangular number $T_n$ is defined as $T_n = 1+2+3+ cdots + n$. What is the greatest possible value of the greatest common divisor of $4T_n$ and $n-1$?
Can someone help me to start this? I have no idea what to do.
number-theory greatest-common-divisor
edited Mar 28 at 21:06
Roddy MacPhee
646118
asked Mar 28 at 17:17
sumisumi
584
$endgroup$
add a comment |
$begingroup$
For all positive integers $n$, the $n$th triangular number $T_n$ is defined as $T_n = 1+2+3+ cdots + n$. What is the greatest possible value of the greatest common divisor of $4T_n$ and $n-1$?
Can someone help me to start this? I have no idea what to do.
number-theory greatest-common-divisor
edited Mar 28 at 21:06
Roddy MacPhee
646118
asked Mar 28 at 17:17
sumisumi
584
$endgroup$
3
$begingroup$
Hint: There is a formula for $T_n$ in terms of n.
$endgroup$
– D.B.
Mar 28 at 17:20
$begingroup$
I think at most $4.$
$endgroup$
– Dbchatto67
Mar 28 at 17:21
$begingroup$
Just plug in some values and you will see a pattern.
$endgroup$
– Infiaria
Mar 28 at 17:21
add a comment |
$begingroup$
For all positive integers $n$, the $n$th triangular number $T_n$ is defined as $T_n = 1+2+3+ cdots + n$. What is the greatest possible value of the greatest common divisor of $4T_n$ and $n-1$?
Can someone help me to start this? I have no idea what to do.
number-theory greatest-common-divisor
edited Mar 28 at 21:06
Roddy MacPhee
646118
asked Mar 28 at 17:17
sumisumi
584
$endgroup$
For all positive integers $n$, the $n$th triangular number $T_n$ is defined as $T_n = 1+2+3+ cdots + n$. What is the greatest possible value of the greatest common divisor of $4T_n$ and $n-1$?
Can someone help me to start this? I have no idea what to do.
number-theory greatest-common-divisor
number-theory greatest-common-divisor
edited Mar 28 at 21:06
Roddy MacPhee
646118
asked Mar 28 at 17:17
sumisumi
584
edited Mar 28 at 21:06
Roddy MacPhee
646118
asked Mar 28 at 17:17
sumisumi
584
edited Mar 28 at 21:06
Roddy MacPhee
646118
edited Mar 28 at 21:06
Roddy MacPhee
646118
edited Mar 28 at 21:06
Roddy MacPhee
646118
646118
asked Mar 28 at 17:17
sumisumi
584
asked Mar 28 at 17:17
sumisumi
584
asked Mar 28 at 17:17
sumisumi
584
584
3
$begingroup$
Hint: There is a formula for $T_n$ in terms of n.
$endgroup$
– D.B.
Mar 28 at 17:20
$begingroup$
I think at most $4.$
$endgroup$
– Dbchatto67
Mar 28 at 17:21
$begingroup$
Just plug in some values and you will see a pattern.
$endgroup$
– Infiaria
Mar 28 at 17:21
add a comment |
3
$begingroup$
Hint: There is a formula for $T_n$ in terms of n.
$endgroup$
– D.B.
Mar 28 at 17:20
$begingroup$
I think at most $4.$
$endgroup$
– Dbchatto67
Mar 28 at 17:21
$begingroup$
Just plug in some values and you will see a pattern.
$endgroup$
– Infiaria
Mar 28 at 17:21
3
3
$begingroup$
Hint: There is a formula for $T_n$ in terms of n.
$endgroup$
– D.B.
Mar 28 at 17:20
$begingroup$
Hint: There is a formula for $T_n$ in terms of n.
$endgroup$
– D.B.
Mar 28 at 17:20
$begingroup$
I think at most $4.$
$endgroup$
– Dbchatto67
Mar 28 at 17:21
$begingroup$
I think at most $4.$
$endgroup$
– Dbchatto67
Mar 28 at 17:21
$begingroup$
Just plug in some values and you will see a pattern.
$endgroup$
– Infiaria
Mar 28 at 17:21
$begingroup$
Just plug in some values and you will see a pattern.
$endgroup$
– Infiaria
Mar 28 at 17:21
add a comment |
3 Answers
3
active
oldest
votes
$begingroup$
Hint: Euclidean division gives $2n(n+1)=(2n+4)(n-1)+4$.
answered Mar 28 at 17:51
lhflhf
167k11172403
$endgroup$
add a comment |
$begingroup$
Can also be done via (additive) Wilson theorem, i.e. pairing up additive inverses mod $,n!-!1,$ shows $,color#c002T_n-1equiv 0pmod!n!-!1,,$ so $,(4T_n,n!-!1)= (4n+color#c004T_n-1,,n!-!1) = (4n,n!-!1) = (4,n!-!1)$
answered Mar 29 at 1:55
Bill DubuqueBill Dubuque
213k29196654
$endgroup$
add a comment |
$begingroup$
$2((n-1)+1)((n-1)+2)=2(n-1)^2+6(n-1)+4$ shows at most 4 can be their gcd.
answered Mar 28 at 20:10
Roddy MacPheeRoddy MacPhee
646118
$endgroup$
add a comment |
Your Answer
StackExchange.ifUsing("editor", function ()
return StackExchange.using("mathjaxEditing", function ()
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix)
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
);
);
, "mathjax-editing");
StackExchange.ready(function()
var channelOptions =
tags: "".split(" "),
id: "69"
;
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function()
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled)
StackExchange.using("snippets", function()
createEditor();
);
else
createEditor();
);
function createEditor()
StackExchange.prepareEditor(
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader:
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
,
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
);
);
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%2f3166160%2ftriangular-numbers-and-gcd%23new-answer', 'question_page');
);
Post as a guest
Required, but never shown
3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Hint: Euclidean division gives $2n(n+1)=(2n+4)(n-1)+4$.
answered Mar 28 at 17:51
lhflhf
167k11172403
$endgroup$
add a comment |
$begingroup$
Hint: Euclidean division gives $2n(n+1)=(2n+4)(n-1)+4$.
answered Mar 28 at 17:51
lhflhf
167k11172403
$endgroup$
add a comment |
$begingroup$
Hint: Euclidean division gives $2n(n+1)=(2n+4)(n-1)+4$.
answered Mar 28 at 17:51
lhflhf
167k11172403
$endgroup$
Hint: Euclidean division gives $2n(n+1)=(2n+4)(n-1)+4$.
answered Mar 28 at 17:51
lhflhf
167k11172403
answered Mar 28 at 17:51
lhflhf
167k11172403
answered Mar 28 at 17:51
lhflhf
167k11172403
answered Mar 28 at 17:51
lhflhf
167k11172403
167k11172403
add a comment |
add a comment |
$begingroup$
Can also be done via (additive) Wilson theorem, i.e. pairing up additive inverses mod $,n!-!1,$ shows $,color#c002T_n-1equiv 0pmod!n!-!1,,$ so $,(4T_n,n!-!1)= (4n+color#c004T_n-1,,n!-!1) = (4n,n!-!1) = (4,n!-!1)$
answered Mar 29 at 1:55
Bill DubuqueBill Dubuque
213k29196654
$endgroup$
add a comment |
$begingroup$
Can also be done via (additive) Wilson theorem, i.e. pairing up additive inverses mod $,n!-!1,$ shows $,color#c002T_n-1equiv 0pmod!n!-!1,,$ so $,(4T_n,n!-!1)= (4n+color#c004T_n-1,,n!-!1) = (4n,n!-!1) = (4,n!-!1)$
answered Mar 29 at 1:55
Bill DubuqueBill Dubuque
213k29196654
$endgroup$
add a comment |
$begingroup$
Can also be done via (additive) Wilson theorem, i.e. pairing up additive inverses mod $,n!-!1,$ shows $,color#c002T_n-1equiv 0pmod!n!-!1,,$ so $,(4T_n,n!-!1)= (4n+color#c004T_n-1,,n!-!1) = (4n,n!-!1) = (4,n!-!1)$
answered Mar 29 at 1:55
Bill DubuqueBill Dubuque
213k29196654
$endgroup$
Can also be done via (additive) Wilson theorem, i.e. pairing up additive inverses mod $,n!-!1,$ shows $,color#c002T_n-1equiv 0pmod!n!-!1,,$ so $,(4T_n,n!-!1)= (4n+color#c004T_n-1,,n!-!1) = (4n,n!-!1) = (4,n!-!1)$
answered Mar 29 at 1:55
Bill DubuqueBill Dubuque
213k29196654
answered Mar 29 at 1:55
Bill DubuqueBill Dubuque
213k29196654
answered Mar 29 at 1:55
Bill DubuqueBill Dubuque
213k29196654
answered Mar 29 at 1:55
Bill DubuqueBill Dubuque
213k29196654
213k29196654
add a comment |
add a comment |
$begingroup$
$2((n-1)+1)((n-1)+2)=2(n-1)^2+6(n-1)+4$ shows at most 4 can be their gcd.
answered Mar 28 at 20:10
Roddy MacPheeRoddy MacPhee
646118
$endgroup$
add a comment |
$begingroup$
$2((n-1)+1)((n-1)+2)=2(n-1)^2+6(n-1)+4$ shows at most 4 can be their gcd.
answered Mar 28 at 20:10
Roddy MacPheeRoddy MacPhee
646118
$endgroup$
add a comment |
$begingroup$
$2((n-1)+1)((n-1)+2)=2(n-1)^2+6(n-1)+4$ shows at most 4 can be their gcd.
answered Mar 28 at 20:10
Roddy MacPheeRoddy MacPhee
646118
$endgroup$
$2((n-1)+1)((n-1)+2)=2(n-1)^2+6(n-1)+4$ shows at most 4 can be their gcd.
answered Mar 28 at 20:10
Roddy MacPheeRoddy MacPhee
646118
answered Mar 28 at 20:10
Roddy MacPheeRoddy MacPhee
646118
answered Mar 28 at 20:10
Roddy MacPheeRoddy MacPhee
646118
answered Mar 28 at 20:10
Roddy MacPheeRoddy MacPhee
646118
646118
add a comment |
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
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%2f3166160%2ftriangular-numbers-and-gcd%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
3
$begingroup$
Hint: There is a formula for $T_n$ in terms of n.
$endgroup$
– D.B.
Mar 28 at 17:20
$begingroup$
I think at most $4.$
$endgroup$
– Dbchatto67
Mar 28 at 17:21
$begingroup$
Just plug in some values and you will see a pattern.
$endgroup$
– Infiaria
Mar 28 at 17:21