What is the smallest number n> 5 so that 5 ^ n ends with “3125”?How to prove that if $aequiv b pmodkn$ then $a^kequiv b^k pmodk^2n$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 $f(x) = 2016$.
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What is the smallest number n> 5 so that 5 ^ n ends with "3125"?
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What is the smallest number n> 5 so that 5 ^ n ends with “3125”?
How to prove that if $aequiv b pmodkn$ then $a^kequiv b^k pmodk^2n$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 $f(x) = 2016$.
$begingroup$
What is the smallest number n> 5 so that 5 ^ n ends with "3125"?
What other examples are there?
calculus
asked 4 hours ago
Catherine Cooper Catherine Cooper
321
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.
$endgroup$
add a comment |
$begingroup$
What is the smallest number n> 5 so that 5 ^ n ends with "3125"?
What other examples are there?
calculus
asked 4 hours ago
Catherine Cooper Catherine Cooper
321
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
4 hours ago
1
$begingroup$
Why not just list them out and find it?
$endgroup$
– Jair Taylor
4 hours ago
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
4 hours ago
$begingroup$
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
55 mins 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
asked 4 hours ago
Catherine Cooper Catherine Cooper
321
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.
$endgroup$
What is the smallest number n> 5 so that 5 ^ n ends with "3125"?
What other examples are there?
calculus
calculus
asked 4 hours ago
Catherine Cooper Catherine Cooper
321
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 4 hours ago
Catherine Cooper Catherine Cooper
321
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 4 hours ago
Catherine Cooper Catherine Cooper
321
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 4 hours ago
Catherine Cooper Catherine Cooper
321
asked 4 hours ago
Catherine Cooper Catherine Cooper
321
321
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.
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.
Catherine Cooper is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
1
$begingroup$
What is your take on this?
$endgroup$
– ADITYA PRAKASH
4 hours ago
1
$begingroup$
Why not just list them out and find it?
$endgroup$
– Jair Taylor
4 hours ago
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
4 hours ago
$begingroup$
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
55 mins ago
add a comment |
1
$begingroup$
What is your take on this?
$endgroup$
– ADITYA PRAKASH
4 hours ago
1
$begingroup$
Why not just list them out and find it?
$endgroup$
– Jair Taylor
4 hours ago
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
4 hours ago
$begingroup$
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
55 mins ago
1
1
$begingroup$
What is your take on this?
$endgroup$
– ADITYA PRAKASH
4 hours ago
$begingroup$
What is your take on this?
$endgroup$
– ADITYA PRAKASH
4 hours ago
1
1
$begingroup$
Why not just list them out and find it?
$endgroup$
– Jair Taylor
4 hours ago
$begingroup$
Why not just list them out and find it?
$endgroup$
– Jair Taylor
4 hours ago
4
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
4 hours ago
$begingroup$
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
4 hours ago
$begingroup$
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
55 mins ago
$begingroup$
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
55 mins ago
add a comment |
4 Answers
4
active
oldest
votes
$begingroup$
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.
answered 4 hours ago
Mostafa AyazMostafa Ayaz
17k3939
$endgroup$
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 $16$?
answered 4 hours ago
Robert IsraelRobert Israel
328k23216469
$endgroup$
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 $3125$.
answered 4 hours ago
Peter ForemanPeter Foreman
3,8371216
$endgroup$
$begingroup$
Why not $5^5 = 3125$.
$endgroup$
– fleablood
4 hours ago
2
$begingroup$
The question states that $ngt5$
$endgroup$
– Peter Foreman
4 hours ago
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 $, beginalign 5, &equiv 1!pmod! color#c004 \ Rightarrow 5^largecolor#c00 4!&equiv 1^largecolor#c00 4!!!! pmod!color#c004^large 2endalign$
answered 3 hours ago
Bill DubuqueBill Dubuque
212k29195654
$endgroup$
add a comment |
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4 Answers
4
active
oldest
votes
4 Answers
4
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
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.
answered 4 hours 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 $n>5$:$$5^nequiv 5^5mod 10000\iff \5^n-4equiv 5mod 16\iff\5^n-5equiv 1mod 16$$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.
answered 4 hours 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 $n>5$:$$5^nequiv 5^5mod 10000\iff \5^n-4equiv 5mod 16\iff\5^n-5equiv 1mod 16$$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.
answered 4 hours ago
Mostafa AyazMostafa Ayaz
17k3939
$endgroup$
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.
answered 4 hours ago
Mostafa AyazMostafa Ayaz
17k3939
answered 4 hours ago
Mostafa AyazMostafa Ayaz
17k3939
answered 4 hours ago
Mostafa AyazMostafa Ayaz
17k3939
answered 4 hours ago
Mostafa AyazMostafa Ayaz
17k3939
17k3939
add a comment |
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 $16$?
answered 4 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 $16$?
answered 4 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 $16$?
answered 4 hours ago
Robert IsraelRobert Israel
328k23216469
$endgroup$
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 $16$?
answered 4 hours ago
Robert IsraelRobert Israel
328k23216469
answered 4 hours ago
Robert IsraelRobert Israel
328k23216469
answered 4 hours ago
Robert IsraelRobert Israel
328k23216469
answered 4 hours ago
Robert IsraelRobert Israel
328k23216469
328k23216469
add a comment |
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 $3125$.
answered 4 hours ago
Peter ForemanPeter Foreman
3,8371216
$endgroup$
$begingroup$
Why not $5^5 = 3125$.
$endgroup$
– fleablood
4 hours ago
2
$begingroup$
The question states that $ngt5$
$endgroup$
– Peter Foreman
4 hours ago
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 $3125$.
answered 4 hours ago
Peter ForemanPeter Foreman
3,8371216
$endgroup$
$begingroup$
Why not $5^5 = 3125$.
$endgroup$
– fleablood
4 hours ago
2
$begingroup$
The question states that $ngt5$
$endgroup$
– Peter Foreman
4 hours ago
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 $3125$.
answered 4 hours ago
Peter ForemanPeter Foreman
3,8371216
$endgroup$
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 $3125$.
answered 4 hours ago
Peter ForemanPeter Foreman
3,8371216
edited 4 hours ago
answered 4 hours ago
Peter ForemanPeter Foreman
3,8371216
answered 4 hours ago
Peter ForemanPeter Foreman
3,8371216
answered 4 hours ago
Peter ForemanPeter Foreman
3,8371216
3,8371216
$begingroup$
Why not $5^5 = 3125$.
$endgroup$
– fleablood
4 hours ago
2
$begingroup$
The question states that $ngt5$
$endgroup$
– Peter Foreman
4 hours ago
add a comment |
$begingroup$
Why not $5^5 = 3125$.
$endgroup$
– fleablood
4 hours ago
2
$begingroup$
The question states that $ngt5$
$endgroup$
– Peter Foreman
4 hours ago
$begingroup$
Why not $5^5 = 3125$.
$endgroup$
– fleablood
4 hours ago
$begingroup$
Why not $5^5 = 3125$.
$endgroup$
– fleablood
4 hours ago
2
2
$begingroup$
The question states that $ngt5$
$endgroup$
– Peter Foreman
4 hours ago
$begingroup$
The question states that $ngt5$
$endgroup$
– Peter Foreman
4 hours ago
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 $, beginalign 5, &equiv 1!pmod! color#c004 \ Rightarrow 5^largecolor#c00 4!&equiv 1^largecolor#c00 4!!!! pmod!color#c004^large 2endalign$
answered 3 hours ago
Bill DubuqueBill Dubuque
212k29195654
$endgroup$
add a comment |
$begingroup$
Hint $, 5^large 5+N! bmod 10^large 4 = 5^large 5(5^largecolor#c00 N! bmod 2^large 4).,$ Now recall $, beginalign 5, &equiv 1!pmod! color#c004 \ Rightarrow 5^largecolor#c00 4!&equiv 1^largecolor#c00 4!!!! pmod!color#c004^large 2endalign$
answered 3 hours ago
Bill DubuqueBill Dubuque
212k29195654
$endgroup$
add a comment |
$begingroup$
Hint $, 5^large 5+N! bmod 10^large 4 = 5^large 5(5^largecolor#c00 N! bmod 2^large 4).,$ Now recall $, beginalign 5, &equiv 1!pmod! color#c004 \ Rightarrow 5^largecolor#c00 4!&equiv 1^largecolor#c00 4!!!! pmod!color#c004^large 2endalign$
answered 3 hours ago
Bill DubuqueBill Dubuque
212k29195654
$endgroup$
Hint $, 5^large 5+N! bmod 10^large 4 = 5^large 5(5^largecolor#c00 N! bmod 2^large 4).,$ Now recall $, beginalign 5, &equiv 1!pmod! color#c004 \ Rightarrow 5^largecolor#c00 4!&equiv 1^largecolor#c00 4!!!! pmod!color#c004^large 2endalign$
answered 3 hours ago
Bill DubuqueBill Dubuque
212k29195654
edited 2 hours ago
answered 3 hours ago
Bill DubuqueBill Dubuque
212k29195654
answered 3 hours ago
Bill DubuqueBill Dubuque
212k29195654
answered 3 hours ago
Bill DubuqueBill Dubuque
212k29195654
212k29195654
add a comment |
add a comment |
Catherine Cooper is a new contributor. Be nice, and check out our Code of Conduct.
Catherine Cooper is a new contributor. Be nice, and check out our Code of Conduct.
Catherine Cooper is a new contributor. Be nice, and check out our Code of Conduct.
Catherine Cooper is a new contributor. Be nice, and check out our Code of Conduct.
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1
$begingroup$
What is your take on this?
$endgroup$
– ADITYA PRAKASH
4 hours ago
1
$begingroup$
Why not just list them out and find it?
$endgroup$
– Jair Taylor
4 hours ago
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
4 hours ago
$begingroup$
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
55 mins ago