Solve the congruence $9x equiv −3 pmod24$. Give your answer as a congruence to the smallest possible modulus, and as a congruence modulo 24.

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I've been able to find the answer as a congruence to the smallest possible modulus (i.e. mod 8) but unsure how to find answer as congruence to mod 24. Also, is everything I've done below correct?:



gcd(9,24) = 3



Therefore, our congruence becomes 3x ≡ -1 (mod 8)



So, 3x ≡ 7 (mod 8)



We must find inverse 'c' of 3 (mod 8), i.e. 3c ≡ 1(mod 8)



gcd(3,8) = 1



let 3c + 8y = 1



Using extended Euclidean Algorithm, we get c = 1



Therefore, solution of 3x ≡ 7 (mod 8) (i.e. smallest possible modulus) is:



x ≡ 7 (mod 8)



Now, how to find solution as a congruence to modulus 24? Assuming everything I've done above is correct.




discrete-mathematics modular-arithmetic




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  • 1




    $9times 7equiv 63equiv 15not equiv -3 pmod 24$.
    – lulu
    Mar 31 '17 at 12:38










  • @lulu Hi lulu, what is this referring to?
    – Programmer
    Mar 31 '17 at 12:40






  • 1




    I am pointing out that your solution is not correct. Even $pmod 8$ it is not correct...$3times 7=21equiv 5not equiv 7pmod 8$.
    – lulu
    Mar 31 '17 at 12:42










  • we get $$5;13;21$$
    – Dr. Sonnhard Graubner
    Mar 31 '17 at 12:54














up vote
2
down vote

favorite












I've been able to find the answer as a congruence to the smallest possible modulus (i.e. mod 8) but unsure how to find answer as congruence to mod 24. Also, is everything I've done below correct?:



gcd(9,24) = 3



Therefore, our congruence becomes 3x ≡ -1 (mod 8)



So, 3x ≡ 7 (mod 8)



We must find inverse 'c' of 3 (mod 8), i.e. 3c ≡ 1(mod 8)



gcd(3,8) = 1



let 3c + 8y = 1



Using extended Euclidean Algorithm, we get c = 1



Therefore, solution of 3x ≡ 7 (mod 8) (i.e. smallest possible modulus) is:



x ≡ 7 (mod 8)



Now, how to find solution as a congruence to modulus 24? Assuming everything I've done above is correct.




discrete-mathematics modular-arithmetic




share|cite|improve this question


















  • 1




    $9times 7equiv 63equiv 15not equiv -3 pmod 24$.
    – lulu
    Mar 31 '17 at 12:38










  • @lulu Hi lulu, what is this referring to?
    – Programmer
    Mar 31 '17 at 12:40






  • 1




    I am pointing out that your solution is not correct. Even $pmod 8$ it is not correct...$3times 7=21equiv 5not equiv 7pmod 8$.
    – lulu
    Mar 31 '17 at 12:42










  • we get $$5;13;21$$
    – Dr. Sonnhard Graubner
    Mar 31 '17 at 12:54












up vote
2
down vote

favorite









up vote
2
down vote

favorite











I've been able to find the answer as a congruence to the smallest possible modulus (i.e. mod 8) but unsure how to find answer as congruence to mod 24. Also, is everything I've done below correct?:



gcd(9,24) = 3



Therefore, our congruence becomes 3x ≡ -1 (mod 8)



So, 3x ≡ 7 (mod 8)



We must find inverse 'c' of 3 (mod 8), i.e. 3c ≡ 1(mod 8)



gcd(3,8) = 1



let 3c + 8y = 1



Using extended Euclidean Algorithm, we get c = 1



Therefore, solution of 3x ≡ 7 (mod 8) (i.e. smallest possible modulus) is:



x ≡ 7 (mod 8)



Now, how to find solution as a congruence to modulus 24? Assuming everything I've done above is correct.




discrete-mathematics modular-arithmetic




share|cite|improve this question














I've been able to find the answer as a congruence to the smallest possible modulus (i.e. mod 8) but unsure how to find answer as congruence to mod 24. Also, is everything I've done below correct?:



gcd(9,24) = 3



Therefore, our congruence becomes 3x ≡ -1 (mod 8)



So, 3x ≡ 7 (mod 8)



We must find inverse 'c' of 3 (mod 8), i.e. 3c ≡ 1(mod 8)



gcd(3,8) = 1



let 3c + 8y = 1



Using extended Euclidean Algorithm, we get c = 1



Therefore, solution of 3x ≡ 7 (mod 8) (i.e. smallest possible modulus) is:



x ≡ 7 (mod 8)



Now, how to find solution as a congruence to modulus 24? Assuming everything I've done above is correct.





discrete-mathematics modular-arithmetic





share|cite|improve this question













share|cite|improve this question




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edited Aug 26 at 13:14









The Pointer

2,7692832




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asked Mar 31 '17 at 12:34









Programmer

366112




366112







  • 1




    $9times 7equiv 63equiv 15not equiv -3 pmod 24$.
    – lulu
    Mar 31 '17 at 12:38










  • @lulu Hi lulu, what is this referring to?
    – Programmer
    Mar 31 '17 at 12:40






  • 1




    I am pointing out that your solution is not correct. Even $pmod 8$ it is not correct...$3times 7=21equiv 5not equiv 7pmod 8$.
    – lulu
    Mar 31 '17 at 12:42










  • we get $$5;13;21$$
    – Dr. Sonnhard Graubner
    Mar 31 '17 at 12:54












  • 1




    $9times 7equiv 63equiv 15not equiv -3 pmod 24$.
    – lulu
    Mar 31 '17 at 12:38










  • @lulu Hi lulu, what is this referring to?
    – Programmer
    Mar 31 '17 at 12:40






  • 1




    I am pointing out that your solution is not correct. Even $pmod 8$ it is not correct...$3times 7=21equiv 5not equiv 7pmod 8$.
    – lulu
    Mar 31 '17 at 12:42










  • we get $$5;13;21$$
    – Dr. Sonnhard Graubner
    Mar 31 '17 at 12:54







1




1




$9times 7equiv 63equiv 15not equiv -3 pmod 24$.
– lulu
Mar 31 '17 at 12:38




$9times 7equiv 63equiv 15not equiv -3 pmod 24$.
– lulu
Mar 31 '17 at 12:38












@lulu Hi lulu, what is this referring to?
– Programmer
Mar 31 '17 at 12:40




@lulu Hi lulu, what is this referring to?
– Programmer
Mar 31 '17 at 12:40




1




1




I am pointing out that your solution is not correct. Even $pmod 8$ it is not correct...$3times 7=21equiv 5not equiv 7pmod 8$.
– lulu
Mar 31 '17 at 12:42




I am pointing out that your solution is not correct. Even $pmod 8$ it is not correct...$3times 7=21equiv 5not equiv 7pmod 8$.
– lulu
Mar 31 '17 at 12:42












we get $$5;13;21$$
– Dr. Sonnhard Graubner
Mar 31 '17 at 12:54




we get $$5;13;21$$
– Dr. Sonnhard Graubner
Mar 31 '17 at 12:54










1 Answer
1






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up vote
2
down vote



accepted










How do you get $c=1$. The inverse of $3c equiv 1 pmod8$ is $c=3$ (since $3times 3=9$). In this way, you obtain $x equiv 5 pmod8$.



Observe that $9 times 5=45$ and $24 times 2=48$, so $x equiv 5 pmod24$.






share|cite|improve this answer




















  • Thanks very much :) I realised my mistake. And also thanks for helping me find mod 24 answer. Appreciate it.
    – Programmer
    Mar 31 '17 at 13:10






  • 1




    Just to say: this is not the $pmod 24$ answer. $xequiv 5 pmod 8$ is correct, $pmod 8$. but then you get $xequiv 5,13,21pmod 24$
    – lulu
    Mar 31 '17 at 13:40











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up vote
2
down vote



accepted










How do you get $c=1$. The inverse of $3c equiv 1 pmod8$ is $c=3$ (since $3times 3=9$). In this way, you obtain $x equiv 5 pmod8$.



Observe that $9 times 5=45$ and $24 times 2=48$, so $x equiv 5 pmod24$.






share|cite|improve this answer




















  • Thanks very much :) I realised my mistake. And also thanks for helping me find mod 24 answer. Appreciate it.
    – Programmer
    Mar 31 '17 at 13:10






  • 1




    Just to say: this is not the $pmod 24$ answer. $xequiv 5 pmod 8$ is correct, $pmod 8$. but then you get $xequiv 5,13,21pmod 24$
    – lulu
    Mar 31 '17 at 13:40















up vote
2
down vote



accepted










How do you get $c=1$. The inverse of $3c equiv 1 pmod8$ is $c=3$ (since $3times 3=9$). In this way, you obtain $x equiv 5 pmod8$.



Observe that $9 times 5=45$ and $24 times 2=48$, so $x equiv 5 pmod24$.






share|cite|improve this answer




















  • Thanks very much :) I realised my mistake. And also thanks for helping me find mod 24 answer. Appreciate it.
    – Programmer
    Mar 31 '17 at 13:10






  • 1




    Just to say: this is not the $pmod 24$ answer. $xequiv 5 pmod 8$ is correct, $pmod 8$. but then you get $xequiv 5,13,21pmod 24$
    – lulu
    Mar 31 '17 at 13:40













up vote
2
down vote



accepted







up vote
2
down vote



accepted






How do you get $c=1$. The inverse of $3c equiv 1 pmod8$ is $c=3$ (since $3times 3=9$). In this way, you obtain $x equiv 5 pmod8$.



Observe that $9 times 5=45$ and $24 times 2=48$, so $x equiv 5 pmod24$.






share|cite|improve this answer












How do you get $c=1$. The inverse of $3c equiv 1 pmod8$ is $c=3$ (since $3times 3=9$). In this way, you obtain $x equiv 5 pmod8$.



Observe that $9 times 5=45$ and $24 times 2=48$, so $x equiv 5 pmod24$.







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Mar 31 '17 at 13:02









TheWanderer

1,76711029




1,76711029











  • Thanks very much :) I realised my mistake. And also thanks for helping me find mod 24 answer. Appreciate it.
    – Programmer
    Mar 31 '17 at 13:10






  • 1




    Just to say: this is not the $pmod 24$ answer. $xequiv 5 pmod 8$ is correct, $pmod 8$. but then you get $xequiv 5,13,21pmod 24$
    – lulu
    Mar 31 '17 at 13:40

















  • Thanks very much :) I realised my mistake. And also thanks for helping me find mod 24 answer. Appreciate it.
    – Programmer
    Mar 31 '17 at 13:10






  • 1




    Just to say: this is not the $pmod 24$ answer. $xequiv 5 pmod 8$ is correct, $pmod 8$. but then you get $xequiv 5,13,21pmod 24$
    – lulu
    Mar 31 '17 at 13:40
















Thanks very much :) I realised my mistake. And also thanks for helping me find mod 24 answer. Appreciate it.
– Programmer
Mar 31 '17 at 13:10




Thanks very much :) I realised my mistake. And also thanks for helping me find mod 24 answer. Appreciate it.
– Programmer
Mar 31 '17 at 13:10




1




1




Just to say: this is not the $pmod 24$ answer. $xequiv 5 pmod 8$ is correct, $pmod 8$. but then you get $xequiv 5,13,21pmod 24$
– lulu
Mar 31 '17 at 13:40





Just to say: this is not the $pmod 24$ answer. $xequiv 5 pmod 8$ is correct, $pmod 8$. but then you get $xequiv 5,13,21pmod 24$
– lulu
Mar 31 '17 at 13:40


















 

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