finding m for which modular arithmetic statement is true
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1
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Suppose for all $ninmathbb Z$, we have $(x + 4n)^2equiv x^2bmod m$. Find all $minmathbb N$ for which
this is a true statement.
I have no idea how to go about finding m. I tried to use the fact that $(x+4n)^2 - x^2$ should be divisible by m, and then used the well defined-ness of $+$ and $×$ operations to deduce something but I failed.
modular-arithmetic
edited Nov 16 at 3:20
Parcly Taxel
41k137198
asked Nov 16 at 1:53
childishsadbino
515
add a comment |
up vote
1
down vote
favorite
Suppose for all $ninmathbb Z$, we have $(x + 4n)^2equiv x^2bmod m$. Find all $minmathbb N$ for which
this is a true statement.
I have no idea how to go about finding m. I tried to use the fact that $(x+4n)^2 - x^2$ should be divisible by m, and then used the well defined-ness of $+$ and $×$ operations to deduce something but I failed.
modular-arithmetic
edited Nov 16 at 3:20
Parcly Taxel
41k137198
asked Nov 16 at 1:53
childishsadbino
515
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Suppose for all $ninmathbb Z$, we have $(x + 4n)^2equiv x^2bmod m$. Find all $minmathbb N$ for which
this is a true statement.
I have no idea how to go about finding m. I tried to use the fact that $(x+4n)^2 - x^2$ should be divisible by m, and then used the well defined-ness of $+$ and $×$ operations to deduce something but I failed.
modular-arithmetic
edited Nov 16 at 3:20
Parcly Taxel
41k137198
asked Nov 16 at 1:53
childishsadbino
515
Suppose for all $ninmathbb Z$, we have $(x + 4n)^2equiv x^2bmod m$. Find all $minmathbb N$ for which
this is a true statement.
I have no idea how to go about finding m. I tried to use the fact that $(x+4n)^2 - x^2$ should be divisible by m, and then used the well defined-ness of $+$ and $×$ operations to deduce something but I failed.
modular-arithmetic
modular-arithmetic
edited Nov 16 at 3:20
Parcly Taxel
41k137198
asked Nov 16 at 1:53
childishsadbino
515
edited Nov 16 at 3:20
Parcly Taxel
41k137198
asked Nov 16 at 1:53
childishsadbino
515
edited Nov 16 at 3:20
Parcly Taxel
41k137198
edited Nov 16 at 3:20
Parcly Taxel
41k137198
edited Nov 16 at 3:20
Parcly Taxel
41k137198
41k137198
asked Nov 16 at 1:53
childishsadbino
515
asked Nov 16 at 1:53
childishsadbino
515
asked Nov 16 at 1:53
childishsadbino
515
515
add a comment |
add a comment |
1 Answer
1
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2
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$$forall n:8nx+16n^2equiv0bmod m$$
Since $x$ is a variable, we must have $mmid8n$ and $mmid16n^2$. We can ignore the second condition as it is implied by the first one. Since $n$ can be any integer, including 1, we must have $mmid8$, so $m=1,2,4,8$.
answered Nov 16 at 3:21
Parcly Taxel
41k137198
OP needs to clarify what $x$ means, e.g. $x$ isn't a "variable" in $,x^pequiv xpmod{p}$.
– Bill Dubuque
Nov 16 at 4:26
@BillDubuque I realize that, but the homework problem I got from my professor didn't mention what x was either, which led to additional confusion. I just assumed x to be fixed and n to be varied, and then repeating the same process with all other x.
– childishsadbino
Nov 16 at 19:19
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
2
down vote
$$forall n:8nx+16n^2equiv0bmod m$$
Since $x$ is a variable, we must have $mmid8n$ and $mmid16n^2$. We can ignore the second condition as it is implied by the first one. Since $n$ can be any integer, including 1, we must have $mmid8$, so $m=1,2,4,8$.
answered Nov 16 at 3:21
Parcly Taxel
41k137198
OP needs to clarify what $x$ means, e.g. $x$ isn't a "variable" in $,x^pequiv xpmod{p}$.
– Bill Dubuque
Nov 16 at 4:26
@BillDubuque I realize that, but the homework problem I got from my professor didn't mention what x was either, which led to additional confusion. I just assumed x to be fixed and n to be varied, and then repeating the same process with all other x.
– childishsadbino
Nov 16 at 19:19
add a comment |
up vote
2
down vote
$$forall n:8nx+16n^2equiv0bmod m$$
Since $x$ is a variable, we must have $mmid8n$ and $mmid16n^2$. We can ignore the second condition as it is implied by the first one. Since $n$ can be any integer, including 1, we must have $mmid8$, so $m=1,2,4,8$.
answered Nov 16 at 3:21
Parcly Taxel
41k137198
OP needs to clarify what $x$ means, e.g. $x$ isn't a "variable" in $,x^pequiv xpmod{p}$.
– Bill Dubuque
Nov 16 at 4:26
@BillDubuque I realize that, but the homework problem I got from my professor didn't mention what x was either, which led to additional confusion. I just assumed x to be fixed and n to be varied, and then repeating the same process with all other x.
– childishsadbino
Nov 16 at 19:19
add a comment |
up vote
2
down vote
up vote
2
down vote
$$forall n:8nx+16n^2equiv0bmod m$$
Since $x$ is a variable, we must have $mmid8n$ and $mmid16n^2$. We can ignore the second condition as it is implied by the first one. Since $n$ can be any integer, including 1, we must have $mmid8$, so $m=1,2,4,8$.
answered Nov 16 at 3:21
Parcly Taxel
41k137198
$$forall n:8nx+16n^2equiv0bmod m$$
Since $x$ is a variable, we must have $mmid8n$ and $mmid16n^2$. We can ignore the second condition as it is implied by the first one. Since $n$ can be any integer, including 1, we must have $mmid8$, so $m=1,2,4,8$.
answered Nov 16 at 3:21
Parcly Taxel
41k137198
answered Nov 16 at 3:21
Parcly Taxel
41k137198
answered Nov 16 at 3:21
Parcly Taxel
41k137198
answered Nov 16 at 3:21
Parcly Taxel
41k137198
41k137198
OP needs to clarify what $x$ means, e.g. $x$ isn't a "variable" in $,x^pequiv xpmod{p}$.
– Bill Dubuque
Nov 16 at 4:26
@BillDubuque I realize that, but the homework problem I got from my professor didn't mention what x was either, which led to additional confusion. I just assumed x to be fixed and n to be varied, and then repeating the same process with all other x.
– childishsadbino
Nov 16 at 19:19
add a comment |
OP needs to clarify what $x$ means, e.g. $x$ isn't a "variable" in $,x^pequiv xpmod{p}$.
– Bill Dubuque
Nov 16 at 4:26
@BillDubuque I realize that, but the homework problem I got from my professor didn't mention what x was either, which led to additional confusion. I just assumed x to be fixed and n to be varied, and then repeating the same process with all other x.
– childishsadbino
Nov 16 at 19:19
OP needs to clarify what $x$ means, e.g. $x$ isn't a "variable" in $,x^pequiv xpmod{p}$.
– Bill Dubuque
Nov 16 at 4:26
OP needs to clarify what $x$ means, e.g. $x$ isn't a "variable" in $,x^pequiv xpmod{p}$.
– Bill Dubuque
Nov 16 at 4:26
@BillDubuque I realize that, but the homework problem I got from my professor didn't mention what x was either, which led to additional confusion. I just assumed x to be fixed and n to be varied, and then repeating the same process with all other x.
– childishsadbino
Nov 16 at 19:19
@BillDubuque I realize that, but the homework problem I got from my professor didn't mention what x was either, which led to additional confusion. I just assumed x to be fixed and n to be varied, and then repeating the same process with all other x.
– childishsadbino
Nov 16 at 19:19
add a comment |
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