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

2

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 Mar 20 at 19:59

Catherine Cooper Catherine Cooper

243

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• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  What is your take on this?
  $endgroup$
  – ADITYA PRAKASH
  Mar 20 at 20:01
  

  
  
  
  


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  1
  

  
  
  
  
  
  

  
  $begingroup$
  Why not just list them out and find it?
  $endgroup$
  – Jair Taylor
  Mar 20 at 20:02
  

  
  
  
  


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  4
  

  
  
  
  
  
  

  
  $begingroup$
  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
  Mar 20 at 20:20
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  The answer to "What is the smallest such n>5?" is easy, so you might as well retitle the question "What are all n>5 such that...?"
  $endgroup$
  – smci
  Mar 20 at 23:55
  
  

  
  
  
  

add a comment | 

2

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 Mar 20 at 19:59

Catherine Cooper Catherine Cooper

243

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  What is your take on this?
  $endgroup$
  – ADITYA PRAKASH
  Mar 20 at 20:01
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  Why not just list them out and find it?
  $endgroup$
  – Jair Taylor
  Mar 20 at 20:02
  

  
  
  
  


• 
  
  
  

  
  4
  

  
  
  
  
  
  

  
  $begingroup$
  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
  Mar 20 at 20:20
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  The answer to "What is the smallest such n>5?" is easy, so you might as well retitle the question "What are all n>5 such that...?"
  $endgroup$
  – smci
  Mar 20 at 23:55
  
  

  
  
  
  

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 Mar 20 at 19:59

Catherine Cooper Catherine Cooper

243

$endgroup$

share|cite|improve this question

asked Mar 20 at 19:59

Catherine Cooper Catherine Cooper

243

share|cite|improve this question

asked Mar 20 at 19:59

Catherine Cooper Catherine Cooper

243

asked Mar 20 at 19:59

Catherine Cooper Catherine Cooper

243

asked Mar 20 at 19:59

Catherine Cooper Catherine Cooper

243

4 Answers
4

active

oldest

votes

5

$begingroup$

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

Note that the following equivalence holds for $n>5$:$$5^nequiv 5^5mod 10000\iff \5^n-4equiv 5mod 16\iff\5^n-5equiv 1mod 16$$Define $mtriangleq n-5ge 1$. Then all the $m$s for which $5^mequiv 1mod 16$ holds are $$m=4kquad,quad kin Bbb N$$this is because $5^4=625equiv 1mod 16$ and therefore $$5^4kequiv5^4k-4equivcdots equiv 5^4equiv 1mod 16$$

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 Mar 20 at 20:17

Mostafa AyazMostafa Ayaz

18.2k31040

$endgroup$

add a comment | 

4

$begingroup$

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 Mar 20 at 20:12

Robert IsraelRobert Israel

330k23218473

$endgroup$

add a comment | 

1

$begingroup$

Well
$$5^9=1953125$$
so the answer is $9$. In fact
$$5^nequiv 5^n-4 mod10^4$$
For $nge 8$, so any value of $5^5+4k$ where $kinmathbbN$ has the last four digits $3125$.

share|cite|improve this answer

edited Mar 20 at 20:16

answered Mar 20 at 20:11

Peter ForemanPeter Foreman

4,9051216

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Why not $5^5 = 3125$.
  $endgroup$
  – fleablood
  Mar 20 at 20:20
  

  
  
  
  


• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  $begingroup$
  The question states that $ngt5$
  $endgroup$
  – Peter Foreman
  Mar 20 at 20:22
  

  
  
  
  

add a comment | 

0

$begingroup$

Hint $, 5^large 5+N! bmod 10^large 4 = 5^large 5(5^largecolor#c00 N! bmod 2^large 4).,$ Now recall $, beginalign 5, &equiv 1!pmod! color#c004 \ Rightarrow 5^largecolor#c00 4!&equiv 1^largecolor#c00 4!!!! pmod!color#c004^large 2endalign$

share|cite|improve this answer

edited Mar 20 at 22:24

answered Mar 20 at 21:34

Bill DubuqueBill Dubuque

213k29196654

$endgroup$

add a comment | 

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5

$begingroup$

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

Note that the following equivalence holds for $n>5$:$$5^nequiv 5^5mod 10000\iff \5^n-4equiv 5mod 16\iff\5^n-5equiv 1mod 16$$Define $mtriangleq n-5ge 1$. Then all the $m$s for which $5^mequiv 1mod 16$ holds are $$m=4kquad,quad kin Bbb N$$this is because $5^4=625equiv 1mod 16$ and therefore $$5^4kequiv5^4k-4equivcdots equiv 5^4equiv 1mod 16$$

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 Mar 20 at 20:17

Mostafa AyazMostafa Ayaz

18.2k31040

$endgroup$

add a comment | 

5

$begingroup$

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

Note that the following equivalence holds for $n>5$:$$5^nequiv 5^5mod 10000\iff \5^n-4equiv 5mod 16\iff\5^n-5equiv 1mod 16$$Define $mtriangleq n-5ge 1$. Then all the $m$s for which $5^mequiv 1mod 16$ holds are $$m=4kquad,quad kin Bbb N$$this is because $5^4=625equiv 1mod 16$ and therefore $$5^4kequiv5^4k-4equivcdots equiv 5^4equiv 1mod 16$$

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 Mar 20 at 20:17

Mostafa AyazMostafa Ayaz

18.2k31040

$endgroup$

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 $n>5$:$$5^nequiv 5^5mod 10000\iff \5^n-4equiv 5mod 16\iff\5^n-5equiv 1mod 16$$Define $mtriangleq n-5ge 1$. Then all the $m$s for which $5^mequiv 1mod 16$ holds are $$m=4kquad,quad kin Bbb N$$this is because $5^4=625equiv 1mod 16$ and therefore $$5^4kequiv5^4k-4equivcdots equiv 5^4equiv 1mod 16$$

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 Mar 20 at 20:17

Mostafa AyazMostafa Ayaz

18.2k31040

$endgroup$

share|cite|improve this answer

answered Mar 20 at 20:17

Mostafa AyazMostafa Ayaz

18.2k31040

answered Mar 20 at 20:17

Mostafa AyazMostafa Ayaz

18.2k31040

answered Mar 20 at 20:17

Mostafa AyazMostafa Ayaz

18.2k31040

4

$begingroup$

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 Mar 20 at 20:12

Robert IsraelRobert Israel

330k23218473

$endgroup$

add a comment | 

4

$begingroup$

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 Mar 20 at 20:12

Robert IsraelRobert Israel

330k23218473

$endgroup$

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 Mar 20 at 20:12

Robert IsraelRobert Israel

330k23218473

$endgroup$

share|cite|improve this answer

answered Mar 20 at 20:12

Robert IsraelRobert Israel

330k23218473

answered Mar 20 at 20:12

Robert IsraelRobert Israel

330k23218473

answered Mar 20 at 20:12

Robert IsraelRobert Israel

330k23218473

1

$begingroup$

Well
$$5^9=1953125$$
so the answer is $9$. In fact
$$5^nequiv 5^n-4 mod10^4$$
For $nge 8$, so any value of $5^5+4k$ where $kinmathbbN$ has the last four digits $3125$.

share|cite|improve this answer

edited Mar 20 at 20:16

answered Mar 20 at 20:11

Peter ForemanPeter Foreman

4,9051216

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Why not $5^5 = 3125$.
  $endgroup$
  – fleablood
  Mar 20 at 20:20
  

  
  
  
  


• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  $begingroup$
  The question states that $ngt5$
  $endgroup$
  – Peter Foreman
  Mar 20 at 20:22
  

  
  
  
  

add a comment | 

1

$begingroup$

Well
$$5^9=1953125$$
so the answer is $9$. In fact
$$5^nequiv 5^n-4 mod10^4$$
For $nge 8$, so any value of $5^5+4k$ where $kinmathbbN$ has the last four digits $3125$.

share|cite|improve this answer

edited Mar 20 at 20:16

answered Mar 20 at 20:11

Peter ForemanPeter Foreman

4,9051216

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Why not $5^5 = 3125$.
  $endgroup$
  – fleablood
  Mar 20 at 20:20
  

  
  
  
  


• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  $begingroup$
  The question states that $ngt5$
  $endgroup$
  – Peter Foreman
  Mar 20 at 20:22
  

  
  
  
  

add a comment | 

$begingroup$

Well
$$5^9=1953125$$
so the answer is $9$. In fact
$$5^nequiv 5^n-4 mod10^4$$
For $nge 8$, so any value of $5^5+4k$ where $kinmathbbN$ has the last four digits $3125$.

share|cite|improve this answer

edited Mar 20 at 20:16

answered Mar 20 at 20:11

Peter ForemanPeter Foreman

4,9051216

$endgroup$

share|cite|improve this answer

edited Mar 20 at 20:16

answered Mar 20 at 20:11

Peter ForemanPeter Foreman

4,9051216

answered Mar 20 at 20:11

Peter ForemanPeter Foreman

4,9051216

answered Mar 20 at 20:11

Peter ForemanPeter Foreman

4,9051216

0

$begingroup$

Hint $, 5^large 5+N! bmod 10^large 4 = 5^large 5(5^largecolor#c00 N! bmod 2^large 4).,$ Now recall $, beginalign 5, &equiv 1!pmod! color#c004 \ Rightarrow 5^largecolor#c00 4!&equiv 1^largecolor#c00 4!!!! pmod!color#c004^large 2endalign$

share|cite|improve this answer

edited Mar 20 at 22:24

answered Mar 20 at 21:34

Bill DubuqueBill Dubuque

213k29196654

$endgroup$

add a comment | 

0

$begingroup$

Hint $, 5^large 5+N! bmod 10^large 4 = 5^large 5(5^largecolor#c00 N! bmod 2^large 4).,$ Now recall $, beginalign 5, &equiv 1!pmod! color#c004 \ Rightarrow 5^largecolor#c00 4!&equiv 1^largecolor#c00 4!!!! pmod!color#c004^large 2endalign$

share|cite|improve this answer

edited Mar 20 at 22:24

answered Mar 20 at 21:34

Bill DubuqueBill Dubuque

213k29196654

$endgroup$

add a comment | 

$begingroup$

Hint $, 5^large 5+N! bmod 10^large 4 = 5^large 5(5^largecolor#c00 N! bmod 2^large 4).,$ Now recall $, beginalign 5, &equiv 1!pmod! color#c004 \ Rightarrow 5^largecolor#c00 4!&equiv 1^largecolor#c00 4!!!! pmod!color#c004^large 2endalign$

share|cite|improve this answer

edited Mar 20 at 22:24

answered Mar 20 at 21:34

Bill DubuqueBill Dubuque

213k29196654

$endgroup$

share|cite|improve this answer

edited Mar 20 at 22:24

answered Mar 20 at 21:34

Bill DubuqueBill Dubuque

213k29196654

answered Mar 20 at 21:34

Bill DubuqueBill Dubuque

213k29196654

answered Mar 20 at 21:34

Bill DubuqueBill Dubuque

213k29196654