Find $438^87493 equiv ~? pmod11$ [closed]How do I compute $a^b,bmod c$ by hand?How to divide $2x equiv 4 pmod 7$ to get just $x equiv Box pmod7$Solving $x^5 equiv 7 mod 13$Infinite exponentiation $n^n^n^…^n equiv m pmod q$ , find m?Modular arithmetic system $x equiv 2 pmod7$ and $x equiv -5 pmod22$How do I find the smallest positive integer $a$ for which $a^n equiv x pmod2^w$?Modular arithmetic problem: $7^x equiv 1 pmod26$Find $6^273 + 8^273pmod49$If $a^2 equiv 4 pmod5$, then $a equiv 2 pmod5$.Solve $x^2 equiv 1 pmod30$ and generalize.Solving $m^3 equiv n^6 pmod19$
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Find $438^87493 equiv ~? pmod11$ [closed]
How do I compute $a^b,bmod c$ by hand?How to divide $2x equiv 4 pmod 7$ to get just $x equiv Box pmod7$Solving $x^5 equiv 7 mod 13$Infinite exponentiation $n^n^n^…^n equiv m pmod q$ , find m?Modular arithmetic system $x equiv 2 pmod7$ and $x equiv -5 pmod22$How do I find the smallest positive integer $a$ for which $a^n equiv x pmod2^w$?Modular arithmetic problem: $7^x equiv 1 pmod26$Find $6^273 + 8^273pmod49$If $a^2 equiv 4 pmod5$, then $a equiv 2 pmod5$.Solve $x^2 equiv 1 pmod30$ and generalize.Solving $m^3 equiv n^6 pmod19$
$begingroup$
How to find the value of '?'
a mod m = b mod m , will this formula be used?
I am taking discrete maths course for CS. And this question is from one of its chapter
modular-arithmetic
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closed as off-topic by José Carlos Santos, Saad, Cesareo, Arnaud D., Javi Mar 21 at 12:46
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – José Carlos Santos, Saad, Cesareo
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$begingroup$
How to find the value of '?'
a mod m = b mod m , will this formula be used?
I am taking discrete maths course for CS. And this question is from one of its chapter
modular-arithmetic
$endgroup$
closed as off-topic by José Carlos Santos, Saad, Cesareo, Arnaud D., Javi Mar 21 at 12:46
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – José Carlos Santos, Saad, Cesareo
2
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What is $438 (mod 11)$? And what is the order of $(Bbb Z / 11 Bbb Z)^*$?
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– Robert Shore
Mar 21 at 6:39
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438 mod 11 would be 9 And I don't know what you mean by order of (Z/11Z)
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– rdr2
Mar 21 at 6:43
$begingroup$
For us to be able to help you effectively, you need to provide more background. Why is this problem of interest to you? How have you tried to solve it?
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– Robert Shore
Mar 21 at 6:44
$begingroup$
I am taking discrete maths course for CS. And this question is from one of its chapter
$endgroup$
– rdr2
Mar 21 at 6:47
1
$begingroup$
Possible duplicate of How do I compute $a^b,bmod c$ by hand?
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– Arnaud D.
Mar 21 at 10:47
|
show 7 more comments
$begingroup$
How to find the value of '?'
a mod m = b mod m , will this formula be used?
I am taking discrete maths course for CS. And this question is from one of its chapter
modular-arithmetic
$endgroup$
How to find the value of '?'
a mod m = b mod m , will this formula be used?
I am taking discrete maths course for CS. And this question is from one of its chapter
modular-arithmetic
modular-arithmetic
edited Mar 21 at 6:51
rdr2
asked Mar 21 at 6:32
rdr2rdr2
12
12
closed as off-topic by José Carlos Santos, Saad, Cesareo, Arnaud D., Javi Mar 21 at 12:46
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – José Carlos Santos, Saad, Cesareo
closed as off-topic by José Carlos Santos, Saad, Cesareo, Arnaud D., Javi Mar 21 at 12:46
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – José Carlos Santos, Saad, Cesareo
2
$begingroup$
What is $438 (mod 11)$? And what is the order of $(Bbb Z / 11 Bbb Z)^*$?
$endgroup$
– Robert Shore
Mar 21 at 6:39
$begingroup$
438 mod 11 would be 9 And I don't know what you mean by order of (Z/11Z)
$endgroup$
– rdr2
Mar 21 at 6:43
$begingroup$
For us to be able to help you effectively, you need to provide more background. Why is this problem of interest to you? How have you tried to solve it?
$endgroup$
– Robert Shore
Mar 21 at 6:44
$begingroup$
I am taking discrete maths course for CS. And this question is from one of its chapter
$endgroup$
– rdr2
Mar 21 at 6:47
1
$begingroup$
Possible duplicate of How do I compute $a^b,bmod c$ by hand?
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– Arnaud D.
Mar 21 at 10:47
|
show 7 more comments
2
$begingroup$
What is $438 (mod 11)$? And what is the order of $(Bbb Z / 11 Bbb Z)^*$?
$endgroup$
– Robert Shore
Mar 21 at 6:39
$begingroup$
438 mod 11 would be 9 And I don't know what you mean by order of (Z/11Z)
$endgroup$
– rdr2
Mar 21 at 6:43
$begingroup$
For us to be able to help you effectively, you need to provide more background. Why is this problem of interest to you? How have you tried to solve it?
$endgroup$
– Robert Shore
Mar 21 at 6:44
$begingroup$
I am taking discrete maths course for CS. And this question is from one of its chapter
$endgroup$
– rdr2
Mar 21 at 6:47
1
$begingroup$
Possible duplicate of How do I compute $a^b,bmod c$ by hand?
$endgroup$
– Arnaud D.
Mar 21 at 10:47
2
2
$begingroup$
What is $438 (mod 11)$? And what is the order of $(Bbb Z / 11 Bbb Z)^*$?
$endgroup$
– Robert Shore
Mar 21 at 6:39
$begingroup$
What is $438 (mod 11)$? And what is the order of $(Bbb Z / 11 Bbb Z)^*$?
$endgroup$
– Robert Shore
Mar 21 at 6:39
$begingroup$
438 mod 11 would be 9 And I don't know what you mean by order of (Z/11Z)
$endgroup$
– rdr2
Mar 21 at 6:43
$begingroup$
438 mod 11 would be 9 And I don't know what you mean by order of (Z/11Z)
$endgroup$
– rdr2
Mar 21 at 6:43
$begingroup$
For us to be able to help you effectively, you need to provide more background. Why is this problem of interest to you? How have you tried to solve it?
$endgroup$
– Robert Shore
Mar 21 at 6:44
$begingroup$
For us to be able to help you effectively, you need to provide more background. Why is this problem of interest to you? How have you tried to solve it?
$endgroup$
– Robert Shore
Mar 21 at 6:44
$begingroup$
I am taking discrete maths course for CS. And this question is from one of its chapter
$endgroup$
– rdr2
Mar 21 at 6:47
$begingroup$
I am taking discrete maths course for CS. And this question is from one of its chapter
$endgroup$
– rdr2
Mar 21 at 6:47
1
1
$begingroup$
Possible duplicate of How do I compute $a^b,bmod c$ by hand?
$endgroup$
– Arnaud D.
Mar 21 at 10:47
$begingroup$
Possible duplicate of How do I compute $a^b,bmod c$ by hand?
$endgroup$
– Arnaud D.
Mar 21 at 10:47
|
show 7 more comments
1 Answer
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Observe that $text gcd (11,438) = 1.$ So by Euler's theorem $438^10 equiv 1 (text mod 11).$ So $438^87490 equiv 1 (text mod 11).$ Also $438 equiv -2 (text mod 11) implies 438^3 equiv -8 equiv 3 (text mod 11).$ Therefore $$438^87493 equiv 3 (text mod 11).$$
If you don't know Euler's theorem or Fermat's little theorem then observe from my above calculation that $438^3 equiv 3 (text mod 11).$ So $438^6 equiv 9 equiv -2 (text mod 11) implies 438^30 equiv (-2)^5 equiv -32 equiv 1 (text mod 11).$
Observe that $87493 = 87480 + 13.$ Since $30 mid 87480$ so $438^87480 equiv 1 (text mod 11).$ Since $438^6 equiv -2 (text mod 11)$ so $438^12 equiv 4 (text mod 11).$ Again $438 equiv -2 (text mod 11).$ So $438^13 equiv 438^12 cdot 438 equiv 4 cdot (-2) equiv -8 equiv 3 (text mod 11).$ Thus we get
$(1)$ $438^87480 equiv 1 (text mod 11).$
$(2)$ $438^13 equiv 3 (text mod 11).$
Therefore what is $438^87493 equiv ~? (text mod 11)$?
$$438^87493 equiv 438^87480+13 equiv 438^87490 cdot 438^13 equiv 1 cdot 3 equiv 3 (text mod 11).$$
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add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Observe that $text gcd (11,438) = 1.$ So by Euler's theorem $438^10 equiv 1 (text mod 11).$ So $438^87490 equiv 1 (text mod 11).$ Also $438 equiv -2 (text mod 11) implies 438^3 equiv -8 equiv 3 (text mod 11).$ Therefore $$438^87493 equiv 3 (text mod 11).$$
If you don't know Euler's theorem or Fermat's little theorem then observe from my above calculation that $438^3 equiv 3 (text mod 11).$ So $438^6 equiv 9 equiv -2 (text mod 11) implies 438^30 equiv (-2)^5 equiv -32 equiv 1 (text mod 11).$
Observe that $87493 = 87480 + 13.$ Since $30 mid 87480$ so $438^87480 equiv 1 (text mod 11).$ Since $438^6 equiv -2 (text mod 11)$ so $438^12 equiv 4 (text mod 11).$ Again $438 equiv -2 (text mod 11).$ So $438^13 equiv 438^12 cdot 438 equiv 4 cdot (-2) equiv -8 equiv 3 (text mod 11).$ Thus we get
$(1)$ $438^87480 equiv 1 (text mod 11).$
$(2)$ $438^13 equiv 3 (text mod 11).$
Therefore what is $438^87493 equiv ~? (text mod 11)$?
$$438^87493 equiv 438^87480+13 equiv 438^87490 cdot 438^13 equiv 1 cdot 3 equiv 3 (text mod 11).$$
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add a comment |
$begingroup$
Observe that $text gcd (11,438) = 1.$ So by Euler's theorem $438^10 equiv 1 (text mod 11).$ So $438^87490 equiv 1 (text mod 11).$ Also $438 equiv -2 (text mod 11) implies 438^3 equiv -8 equiv 3 (text mod 11).$ Therefore $$438^87493 equiv 3 (text mod 11).$$
If you don't know Euler's theorem or Fermat's little theorem then observe from my above calculation that $438^3 equiv 3 (text mod 11).$ So $438^6 equiv 9 equiv -2 (text mod 11) implies 438^30 equiv (-2)^5 equiv -32 equiv 1 (text mod 11).$
Observe that $87493 = 87480 + 13.$ Since $30 mid 87480$ so $438^87480 equiv 1 (text mod 11).$ Since $438^6 equiv -2 (text mod 11)$ so $438^12 equiv 4 (text mod 11).$ Again $438 equiv -2 (text mod 11).$ So $438^13 equiv 438^12 cdot 438 equiv 4 cdot (-2) equiv -8 equiv 3 (text mod 11).$ Thus we get
$(1)$ $438^87480 equiv 1 (text mod 11).$
$(2)$ $438^13 equiv 3 (text mod 11).$
Therefore what is $438^87493 equiv ~? (text mod 11)$?
$$438^87493 equiv 438^87480+13 equiv 438^87490 cdot 438^13 equiv 1 cdot 3 equiv 3 (text mod 11).$$
$endgroup$
add a comment |
$begingroup$
Observe that $text gcd (11,438) = 1.$ So by Euler's theorem $438^10 equiv 1 (text mod 11).$ So $438^87490 equiv 1 (text mod 11).$ Also $438 equiv -2 (text mod 11) implies 438^3 equiv -8 equiv 3 (text mod 11).$ Therefore $$438^87493 equiv 3 (text mod 11).$$
If you don't know Euler's theorem or Fermat's little theorem then observe from my above calculation that $438^3 equiv 3 (text mod 11).$ So $438^6 equiv 9 equiv -2 (text mod 11) implies 438^30 equiv (-2)^5 equiv -32 equiv 1 (text mod 11).$
Observe that $87493 = 87480 + 13.$ Since $30 mid 87480$ so $438^87480 equiv 1 (text mod 11).$ Since $438^6 equiv -2 (text mod 11)$ so $438^12 equiv 4 (text mod 11).$ Again $438 equiv -2 (text mod 11).$ So $438^13 equiv 438^12 cdot 438 equiv 4 cdot (-2) equiv -8 equiv 3 (text mod 11).$ Thus we get
$(1)$ $438^87480 equiv 1 (text mod 11).$
$(2)$ $438^13 equiv 3 (text mod 11).$
Therefore what is $438^87493 equiv ~? (text mod 11)$?
$$438^87493 equiv 438^87480+13 equiv 438^87490 cdot 438^13 equiv 1 cdot 3 equiv 3 (text mod 11).$$
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Observe that $text gcd (11,438) = 1.$ So by Euler's theorem $438^10 equiv 1 (text mod 11).$ So $438^87490 equiv 1 (text mod 11).$ Also $438 equiv -2 (text mod 11) implies 438^3 equiv -8 equiv 3 (text mod 11).$ Therefore $$438^87493 equiv 3 (text mod 11).$$
If you don't know Euler's theorem or Fermat's little theorem then observe from my above calculation that $438^3 equiv 3 (text mod 11).$ So $438^6 equiv 9 equiv -2 (text mod 11) implies 438^30 equiv (-2)^5 equiv -32 equiv 1 (text mod 11).$
Observe that $87493 = 87480 + 13.$ Since $30 mid 87480$ so $438^87480 equiv 1 (text mod 11).$ Since $438^6 equiv -2 (text mod 11)$ so $438^12 equiv 4 (text mod 11).$ Again $438 equiv -2 (text mod 11).$ So $438^13 equiv 438^12 cdot 438 equiv 4 cdot (-2) equiv -8 equiv 3 (text mod 11).$ Thus we get
$(1)$ $438^87480 equiv 1 (text mod 11).$
$(2)$ $438^13 equiv 3 (text mod 11).$
Therefore what is $438^87493 equiv ~? (text mod 11)$?
$$438^87493 equiv 438^87480+13 equiv 438^87490 cdot 438^13 equiv 1 cdot 3 equiv 3 (text mod 11).$$
edited Mar 21 at 8:03
answered Mar 21 at 6:45
Dbchatto67Dbchatto67
2,445522
2,445522
add a comment |
add a comment |
2
$begingroup$
What is $438 (mod 11)$? And what is the order of $(Bbb Z / 11 Bbb Z)^*$?
$endgroup$
– Robert Shore
Mar 21 at 6:39
$begingroup$
438 mod 11 would be 9 And I don't know what you mean by order of (Z/11Z)
$endgroup$
– rdr2
Mar 21 at 6:43
$begingroup$
For us to be able to help you effectively, you need to provide more background. Why is this problem of interest to you? How have you tried to solve it?
$endgroup$
– Robert Shore
Mar 21 at 6:44
$begingroup$
I am taking discrete maths course for CS. And this question is from one of its chapter
$endgroup$
– rdr2
Mar 21 at 6:47
1
$begingroup$
Possible duplicate of How do I compute $a^b,bmod c$ by hand?
$endgroup$
– Arnaud D.
Mar 21 at 10:47