Sfrgttk

What's the name of the logical fallacy where a debater extends a statement far beyond the original statement to make it true?

Why do Radio Buttons not fill the entire outer circle?

Can I cause damage to electrical appliances by unplugging them when they are turned on?

Limit max CPU usage SQL SERVER with WSRM

How to get directions in deep space?

Telemetry for feature health

How to test the sharpness of a knife?

Identifying "long and narrow" polygons in with Postgis

Did I make a mistake by ccing email to boss to others?

Review your own paper in Mathematics

Do I have to know the General Relativity theory to understand the concept of inertial frame?

Does Doodling or Improvising on the Piano Have Any Benefits?

I'm just a whisper. Who am I?

How would a solely written language work mechanically

Difference between shutdown options

Has the laser at Magurele, Romania reached a tenth of the Sun's power?

How much do grades matter for a future academia position?

What is this high flying aircraft over Pennsylvania?

What happens if a creature's ETB would bounce Thalia, Heretic Cathar?

"Oh no!" in Latin

What happens if I try to grapple mirror image?

Is there anyway, I can have two passwords for my wi-fi

Quoting Keynes in a lecture

When and why was runway 07/25 at Kai Tak removed?

What is the smallest number n> 5 so that 5 ^ n ends with “3125”?

How to prove that if $aequiv b pmod{kn}$ then $a^kequiv b^k pmod{k^2n}$Horizontal tank with hemispherical ends depth to capacity calculationDoes the smallest real number that satisfies $2^xge bx$ have logarithmic order?Determine the smallest number POptimization, find the dimensions of the poster with the smallest areaIs $s(t) = 1/(1+t^2)$ a bounded function? If so, find the smallest $M$Continous function approximating the precision of a number.What is the smallest value of this sequence?Find the smallest real number $agt 0$ for which the equation $a^x=x$ has no real solutionsGiven a point A (3,4) What is the smallest segment passing through A and makes a right triangle with the coordinates$f(n) =$ the smallest prime factor of $n$. Prove that the number of solutions to the equation $f(x) = 2016$.

1

$begingroup$

What is the smallest number n> 5 so that 5 ^ n ends with "3125"?

What other examples are there?

calculus

share|cite|improve this question

asked 2 hours ago

Catherine Cooper Catherine Cooper

291

New contributor

Catherine Cooper is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  What is your take on this?
  $endgroup$
  – ADITYA PRAKASH
  2 hours ago
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  Why not just list them out and find it?
  $endgroup$
  – Jair Taylor
  2 hours ago
  

  
  
  
  


• 
  
  
  

  
  3
  

  
  
  
  
  
  

  
  $begingroup$
  Why not just do it? It's not $1$ because $5^1=5$. It's not $2$ because $5^2 = 25$. What's to keep you from just continuing?
  $endgroup$
  – fleablood
  1 hour ago
  

  
  
  
  

add a comment | 

1

$begingroup$

What is the smallest number n> 5 so that 5 ^ n ends with "3125"?

What other examples are there?

calculus

share|cite|improve this question

asked 2 hours ago

Catherine Cooper Catherine Cooper

291

New contributor

Catherine Cooper is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  What is your take on this?
  $endgroup$
  – ADITYA PRAKASH
  2 hours ago
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  Why not just list them out and find it?
  $endgroup$
  – Jair Taylor
  2 hours ago
  

  
  
  
  


• 
  
  
  

  
  3
  

  
  
  
  
  
  

  
  $begingroup$
  Why not just do it? It's not $1$ because $5^1=5$. It's not $2$ because $5^2 = 25$. What's to keep you from just continuing?
  $endgroup$
  – fleablood
  1 hour ago
  

  
  
  
  

add a comment | 

$begingroup$

What is the smallest number n> 5 so that 5 ^ n ends with "3125"?

What other examples are there?

calculus

share|cite|improve this question

asked 2 hours ago

Catherine Cooper Catherine Cooper

291

New contributor

Catherine Cooper is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

$endgroup$

share|cite|improve this question

asked 2 hours ago

Catherine Cooper Catherine Cooper

291

New contributor

Catherine Cooper is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

share|cite|improve this question

asked 2 hours ago

Catherine Cooper Catherine Cooper

291

New contributor

Catherine Cooper is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

asked 2 hours ago

Catherine Cooper Catherine Cooper

291

New contributor

Catherine Cooper is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

asked 2 hours ago

Catherine Cooper Catherine Cooper

291

4 Answers
4

active

oldest

votes

3

$begingroup$

So, we are looking for all $n>5$ for which $5^nequiv 3125=5^5mod 10000$.

Note that the following equivalence holds for $n>5$:$$5^nequiv 5^5mod 10000\iff \5^{n-4}equiv 5mod 16\iff\5^{n-5}equiv 1mod 16$$Define $mtriangleq n-5ge 1$. Then all the $m$s for which $5^{m}equiv 1mod 16$ holds are $$m=4kquad,quad kin Bbb N$$this is because $5^4=625equiv 1mod 16$ and therefore $$5^{4k}equiv5^{4k-4}equivcdots equiv 5^{4}equiv 1mod 16$$

Conclusion

All $n>5$s for which $5^n$ ends up with $3125$ can be found from $$n=4k+5quad,quad kin Bbb N$$ and the smallest such $n$ is 9.

share|cite|improve this answer

answered 1 hour ago

Mostafa AyazMostafa Ayaz

17k3939

$endgroup$

add a comment | 

2

$begingroup$

Hint: $5^n equiv 5^5 mod 10^4$ if and only if $5^n equiv 5^5 mod 2^4$. What is the multiplicative order of $5$ mod $16$?

share|cite|improve this answer

answered 2 hours ago

Robert IsraelRobert Israel

328k23216469

$endgroup$

add a comment | 

0

$begingroup$

Well
$$5^9=1953125$$
so the answer is $9$. In fact
$$5^nequiv 5^{n-4} mod{10^4}$$
For $nge 8$, so any value of $5^{5+4k}$ where $kinmathbb{N}$ has the last four digits $3125$.

share|cite|improve this answer

edited 1 hour ago

answered 2 hours ago

Peter ForemanPeter Foreman

3,8371216

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Why not $5^5 = 3125$.
  $endgroup$
  – fleablood
  1 hour ago
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  The question states that $ngt5$
  $endgroup$
  – Peter Foreman
  1 hour ago
  

  
  
  
  

add a comment | 

0

$begingroup$

Hint $ 5^{large 5+N}! bmod 10^{large 4} = 5^{large 5}(5^{largecolor{#c00} N}! bmod 2^{large 4}),,$ and $ 5^{largecolor{#c00} 4}equiv 1^{largecolor{#c00} 4}! pmod{!4^{large 2}},$ by $,5 equiv 1pmod{! color{#c00}4} $

share|cite|improve this answer

answered 38 mins ago

Bill DubuqueBill Dubuque

212k29195654

$endgroup$

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
});


}
});

Catherine Cooper is a new contributor. Be nice, and check out our Code of Conduct.

draft saved

draft discarded

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

Name

Email

Required, but never shown

StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3155943%2fwhat-is-the-smallest-number-n-5-so-that-5-n-ends-with-3125%23new-answer', 'question_page');
}
);

Post as a guest

Name

Email

Required, but never shown

3

$begingroup$

So, we are looking for all $n>5$ for which $5^nequiv 3125=5^5mod 10000$.

Note that the following equivalence holds for $n>5$:$$5^nequiv 5^5mod 10000\iff \5^{n-4}equiv 5mod 16\iff\5^{n-5}equiv 1mod 16$$Define $mtriangleq n-5ge 1$. Then all the $m$s for which $5^{m}equiv 1mod 16$ holds are $$m=4kquad,quad kin Bbb N$$this is because $5^4=625equiv 1mod 16$ and therefore $$5^{4k}equiv5^{4k-4}equivcdots equiv 5^{4}equiv 1mod 16$$

Conclusion

All $n>5$s for which $5^n$ ends up with $3125$ can be found from $$n=4k+5quad,quad kin Bbb N$$ and the smallest such $n$ is 9.

share|cite|improve this answer

answered 1 hour ago

Mostafa AyazMostafa Ayaz

17k3939

$endgroup$

add a comment | 

3

$begingroup$

So, we are looking for all $n>5$ for which $5^nequiv 3125=5^5mod 10000$.

Note that the following equivalence holds for $n>5$:$$5^nequiv 5^5mod 10000\iff \5^{n-4}equiv 5mod 16\iff\5^{n-5}equiv 1mod 16$$Define $mtriangleq n-5ge 1$. Then all the $m$s for which $5^{m}equiv 1mod 16$ holds are $$m=4kquad,quad kin Bbb N$$this is because $5^4=625equiv 1mod 16$ and therefore $$5^{4k}equiv5^{4k-4}equivcdots equiv 5^{4}equiv 1mod 16$$

Conclusion

All $n>5$s for which $5^n$ ends up with $3125$ can be found from $$n=4k+5quad,quad kin Bbb N$$ and the smallest such $n$ is 9.

share|cite|improve this answer

answered 1 hour ago

Mostafa AyazMostafa Ayaz

17k3939

$endgroup$

add a comment | 

$begingroup$

So, we are looking for all $n>5$ for which $5^nequiv 3125=5^5mod 10000$.

Note that the following equivalence holds for $n>5$:$$5^nequiv 5^5mod 10000\iff \5^{n-4}equiv 5mod 16\iff\5^{n-5}equiv 1mod 16$$Define $mtriangleq n-5ge 1$. Then all the $m$s for which $5^{m}equiv 1mod 16$ holds are $$m=4kquad,quad kin Bbb N$$this is because $5^4=625equiv 1mod 16$ and therefore $$5^{4k}equiv5^{4k-4}equivcdots equiv 5^{4}equiv 1mod 16$$

Conclusion

All $n>5$s for which $5^n$ ends up with $3125$ can be found from $$n=4k+5quad,quad kin Bbb N$$ and the smallest such $n$ is 9.

share|cite|improve this answer

answered 1 hour ago

Mostafa AyazMostafa Ayaz

17k3939

$endgroup$

share|cite|improve this answer

answered 1 hour ago

Mostafa AyazMostafa Ayaz

17k3939

answered 1 hour ago

Mostafa AyazMostafa Ayaz

17k3939

answered 1 hour ago

Mostafa AyazMostafa Ayaz

17k3939

2

$begingroup$

Hint: $5^n equiv 5^5 mod 10^4$ if and only if $5^n equiv 5^5 mod 2^4$. What is the multiplicative order of $5$ mod $16$?

share|cite|improve this answer

answered 2 hours ago

Robert IsraelRobert Israel

328k23216469

$endgroup$

add a comment | 

2

$begingroup$

Hint: $5^n equiv 5^5 mod 10^4$ if and only if $5^n equiv 5^5 mod 2^4$. What is the multiplicative order of $5$ mod $16$?

share|cite|improve this answer

answered 2 hours ago

Robert IsraelRobert Israel

328k23216469

$endgroup$

add a comment | 

$begingroup$

Hint: $5^n equiv 5^5 mod 10^4$ if and only if $5^n equiv 5^5 mod 2^4$. What is the multiplicative order of $5$ mod $16$?

share|cite|improve this answer

answered 2 hours ago

Robert IsraelRobert Israel

328k23216469

$endgroup$

share|cite|improve this answer

answered 2 hours ago

Robert IsraelRobert Israel

328k23216469

answered 2 hours ago

Robert IsraelRobert Israel

328k23216469

answered 2 hours ago

Robert IsraelRobert Israel

328k23216469

0

$begingroup$

Well
$$5^9=1953125$$
so the answer is $9$. In fact
$$5^nequiv 5^{n-4} mod{10^4}$$
For $nge 8$, so any value of $5^{5+4k}$ where $kinmathbb{N}$ has the last four digits $3125$.

share|cite|improve this answer

edited 1 hour ago

answered 2 hours ago

Peter ForemanPeter Foreman

3,8371216

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Why not $5^5 = 3125$.
  $endgroup$
  – fleablood
  1 hour ago
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  The question states that $ngt5$
  $endgroup$
  – Peter Foreman
  1 hour ago
  

  
  
  
  

add a comment | 

0

$begingroup$

Well
$$5^9=1953125$$
so the answer is $9$. In fact
$$5^nequiv 5^{n-4} mod{10^4}$$
For $nge 8$, so any value of $5^{5+4k}$ where $kinmathbb{N}$ has the last four digits $3125$.

share|cite|improve this answer

edited 1 hour ago

answered 2 hours ago

Peter ForemanPeter Foreman

3,8371216

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Why not $5^5 = 3125$.
  $endgroup$
  – fleablood
  1 hour ago
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  The question states that $ngt5$
  $endgroup$
  – Peter Foreman
  1 hour ago
  

  
  
  
  

add a comment | 

$begingroup$

Well
$$5^9=1953125$$
so the answer is $9$. In fact
$$5^nequiv 5^{n-4} mod{10^4}$$
For $nge 8$, so any value of $5^{5+4k}$ where $kinmathbb{N}$ has the last four digits $3125$.

share|cite|improve this answer

edited 1 hour ago

answered 2 hours ago

Peter ForemanPeter Foreman

3,8371216

$endgroup$

share|cite|improve this answer

edited 1 hour ago

answered 2 hours ago

Peter ForemanPeter Foreman

3,8371216

answered 2 hours ago

Peter ForemanPeter Foreman

3,8371216

answered 2 hours ago

Peter ForemanPeter Foreman

3,8371216

0

$begingroup$

Hint $ 5^{large 5+N}! bmod 10^{large 4} = 5^{large 5}(5^{largecolor{#c00} N}! bmod 2^{large 4}),,$ and $ 5^{largecolor{#c00} 4}equiv 1^{largecolor{#c00} 4}! pmod{!4^{large 2}},$ by $,5 equiv 1pmod{! color{#c00}4} $

share|cite|improve this answer

answered 38 mins ago

Bill DubuqueBill Dubuque

212k29195654

$endgroup$

add a comment | 

0

$begingroup$

Hint $ 5^{large 5+N}! bmod 10^{large 4} = 5^{large 5}(5^{largecolor{#c00} N}! bmod 2^{large 4}),,$ and $ 5^{largecolor{#c00} 4}equiv 1^{largecolor{#c00} 4}! pmod{!4^{large 2}},$ by $,5 equiv 1pmod{! color{#c00}4} $

share|cite|improve this answer

answered 38 mins ago

Bill DubuqueBill Dubuque

212k29195654

$endgroup$

add a comment | 

$begingroup$

Hint $ 5^{large 5+N}! bmod 10^{large 4} = 5^{large 5}(5^{largecolor{#c00} N}! bmod 2^{large 4}),,$ and $ 5^{largecolor{#c00} 4}equiv 1^{largecolor{#c00} 4}! pmod{!4^{large 2}},$ by $,5 equiv 1pmod{! color{#c00}4} $

share|cite|improve this answer

answered 38 mins ago

Bill DubuqueBill Dubuque

212k29195654

$endgroup$

share|cite|improve this answer

answered 38 mins ago

Bill DubuqueBill Dubuque

212k29195654

answered 38 mins ago

Bill DubuqueBill Dubuque

212k29195654

answered 38 mins ago

Bill DubuqueBill Dubuque

212k29195654