Solving the Congruence $20x equiv 16 pmod92$ and Giving Answer As a Congruence to the Smallest Possible Modulus
Clash Royale CLAN TAG#URR8PPP
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I have the following problem:
Solve the congruence $20x equiv 16 pmod92$. Give your answer as
(i) a congruence to the smallest possible modulus;
(ii) a congruence modulo $92$.
I just recently solved another congruence equations problem:
Solve the following congruences, or explain why they have no solution:
(i) $28x equiv 3 pmod67$;
(ii) $29x equiv 3 pmod67$.
I'm confused about how to solve (i) for the first problem. I usually solve these problems by using (1) the Euclidean algorithm to find the greatest common divisor, and (2) then using the extended Euclidean algorithm. But how would the way you solve (i) in the first problem differ from how you solve the second problem?
Also, would I be correct in saying that solving (ii) of the first problem is just done in the same way you solve the second problem?
I found this related question on congruences to the smallest possible modulus, but it doesn't seem to actually explain anything; it just goes straight to some calculations.
I would greatly appreciate it if people could please take the time to clarify this.
modular-arithmetic
asked Aug 26 at 6:58
The Pointer
2,7692832
add a comment |Â
up vote
0
down vote
favorite
I have the following problem:
Solve the congruence $20x equiv 16 pmod92$. Give your answer as
(i) a congruence to the smallest possible modulus;
(ii) a congruence modulo $92$.
I just recently solved another congruence equations problem:
Solve the following congruences, or explain why they have no solution:
(i) $28x equiv 3 pmod67$;
(ii) $29x equiv 3 pmod67$.
I'm confused about how to solve (i) for the first problem. I usually solve these problems by using (1) the Euclidean algorithm to find the greatest common divisor, and (2) then using the extended Euclidean algorithm. But how would the way you solve (i) in the first problem differ from how you solve the second problem?
Also, would I be correct in saying that solving (ii) of the first problem is just done in the same way you solve the second problem?
I found this related question on congruences to the smallest possible modulus, but it doesn't seem to actually explain anything; it just goes straight to some calculations.
I would greatly appreciate it if people could please take the time to clarify this.
modular-arithmetic
asked Aug 26 at 6:58
The Pointer
2,7692832
1
So, what is the GCD in the first example?
– Lord Shark the Unknown
Aug 26 at 7:11
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I have the following problem:
Solve the congruence $20x equiv 16 pmod92$. Give your answer as
(i) a congruence to the smallest possible modulus;
(ii) a congruence modulo $92$.
I just recently solved another congruence equations problem:
Solve the following congruences, or explain why they have no solution:
(i) $28x equiv 3 pmod67$;
(ii) $29x equiv 3 pmod67$.
I'm confused about how to solve (i) for the first problem. I usually solve these problems by using (1) the Euclidean algorithm to find the greatest common divisor, and (2) then using the extended Euclidean algorithm. But how would the way you solve (i) in the first problem differ from how you solve the second problem?
Also, would I be correct in saying that solving (ii) of the first problem is just done in the same way you solve the second problem?
I found this related question on congruences to the smallest possible modulus, but it doesn't seem to actually explain anything; it just goes straight to some calculations.
I would greatly appreciate it if people could please take the time to clarify this.
modular-arithmetic
asked Aug 26 at 6:58
The Pointer
2,7692832
I have the following problem:
Solve the congruence $20x equiv 16 pmod92$. Give your answer as
(i) a congruence to the smallest possible modulus;
(ii) a congruence modulo $92$.
I just recently solved another congruence equations problem:
Solve the following congruences, or explain why they have no solution:
(i) $28x equiv 3 pmod67$;
(ii) $29x equiv 3 pmod67$.
I'm confused about how to solve (i) for the first problem. I usually solve these problems by using (1) the Euclidean algorithm to find the greatest common divisor, and (2) then using the extended Euclidean algorithm. But how would the way you solve (i) in the first problem differ from how you solve the second problem?
Also, would I be correct in saying that solving (ii) of the first problem is just done in the same way you solve the second problem?
I found this related question on congruences to the smallest possible modulus, but it doesn't seem to actually explain anything; it just goes straight to some calculations.
I would greatly appreciate it if people could please take the time to clarify this.
modular-arithmetic
asked Aug 26 at 6:58
The Pointer
2,7692832
edited Aug 26 at 7:13
asked Aug 26 at 6:58
The Pointer
2,7692832
asked Aug 26 at 6:58
The Pointer
2,7692832
asked Aug 26 at 6:58
The Pointer
2,7692832
2,7692832
1
So, what is the GCD in the first example?
– Lord Shark the Unknown
Aug 26 at 7:11
add a comment |Â
1
So, what is the GCD in the first example?
– Lord Shark the Unknown
Aug 26 at 7:11
1
1
So, what is the GCD in the first example?
– Lord Shark the Unknown
Aug 26 at 7:11
So, what is the GCD in the first example?
– Lord Shark the Unknown
Aug 26 at 7:11
add a comment |Â
1 Answer
1
active
oldest
votes
up vote
1
down vote
accepted
Yor first equation can by written as $$5xequiv 4mod 23$$, then you can write
$$xequiv frac45equiv frac275equiv frac505equiv 10mod 23$$
answered Aug 26 at 7:12
Dr. Sonnhard Graubner
67.8k32660
Oh, so that's what is meant by smallest possible modulus: we actually have to use the laws of acceptable algebra on congruence equations to get the modulus to the smallest possible integer, and then we solve it as normal?
– The Pointer
Aug 26 at 7:16
1
Yes this is right, have you got the solutions for the other equations?
– Dr. Sonnhard Graubner
Aug 26 at 7:17
Yes, I already solved them. Thanks for the clarification!
– The Pointer
Aug 26 at 7:18
1
Ok, nice that i could help you, have a nice day!
– Dr. Sonnhard Graubner
Aug 26 at 7:18
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
accepted
Yor first equation can by written as $$5xequiv 4mod 23$$, then you can write
$$xequiv frac45equiv frac275equiv frac505equiv 10mod 23$$
answered Aug 26 at 7:12
Dr. Sonnhard Graubner
67.8k32660
Oh, so that's what is meant by smallest possible modulus: we actually have to use the laws of acceptable algebra on congruence equations to get the modulus to the smallest possible integer, and then we solve it as normal?
– The Pointer
Aug 26 at 7:16
1
Yes this is right, have you got the solutions for the other equations?
– Dr. Sonnhard Graubner
Aug 26 at 7:17
Yes, I already solved them. Thanks for the clarification!
– The Pointer
Aug 26 at 7:18
1
Ok, nice that i could help you, have a nice day!
– Dr. Sonnhard Graubner
Aug 26 at 7:18
add a comment |Â
up vote
1
down vote
accepted
Yor first equation can by written as $$5xequiv 4mod 23$$, then you can write
$$xequiv frac45equiv frac275equiv frac505equiv 10mod 23$$
answered Aug 26 at 7:12
Dr. Sonnhard Graubner
67.8k32660
Oh, so that's what is meant by smallest possible modulus: we actually have to use the laws of acceptable algebra on congruence equations to get the modulus to the smallest possible integer, and then we solve it as normal?
– The Pointer
Aug 26 at 7:16
1
Yes this is right, have you got the solutions for the other equations?
– Dr. Sonnhard Graubner
Aug 26 at 7:17
Yes, I already solved them. Thanks for the clarification!
– The Pointer
Aug 26 at 7:18
1
Ok, nice that i could help you, have a nice day!
– Dr. Sonnhard Graubner
Aug 26 at 7:18
add a comment |Â
up vote
1
down vote
accepted
up vote
1
down vote
accepted
Yor first equation can by written as $$5xequiv 4mod 23$$, then you can write
$$xequiv frac45equiv frac275equiv frac505equiv 10mod 23$$
answered Aug 26 at 7:12
Dr. Sonnhard Graubner
67.8k32660
Yor first equation can by written as $$5xequiv 4mod 23$$, then you can write
$$xequiv frac45equiv frac275equiv frac505equiv 10mod 23$$
answered Aug 26 at 7:12
Dr. Sonnhard Graubner
67.8k32660
answered Aug 26 at 7:12
Dr. Sonnhard Graubner
67.8k32660
answered Aug 26 at 7:12
Dr. Sonnhard Graubner
67.8k32660
answered Aug 26 at 7:12
Dr. Sonnhard Graubner
67.8k32660
67.8k32660
Oh, so that's what is meant by smallest possible modulus: we actually have to use the laws of acceptable algebra on congruence equations to get the modulus to the smallest possible integer, and then we solve it as normal?
– The Pointer
Aug 26 at 7:16
1
Yes this is right, have you got the solutions for the other equations?
– Dr. Sonnhard Graubner
Aug 26 at 7:17
Yes, I already solved them. Thanks for the clarification!
– The Pointer
Aug 26 at 7:18
1
Ok, nice that i could help you, have a nice day!
– Dr. Sonnhard Graubner
Aug 26 at 7:18
add a comment |Â
Oh, so that's what is meant by smallest possible modulus: we actually have to use the laws of acceptable algebra on congruence equations to get the modulus to the smallest possible integer, and then we solve it as normal?
– The Pointer
Aug 26 at 7:16
1
Yes this is right, have you got the solutions for the other equations?
– Dr. Sonnhard Graubner
Aug 26 at 7:17
Yes, I already solved them. Thanks for the clarification!
– The Pointer
Aug 26 at 7:18
1
Ok, nice that i could help you, have a nice day!
– Dr. Sonnhard Graubner
Aug 26 at 7:18
Oh, so that's what is meant by smallest possible modulus: we actually have to use the laws of acceptable algebra on congruence equations to get the modulus to the smallest possible integer, and then we solve it as normal?
– The Pointer
Aug 26 at 7:16
Oh, so that's what is meant by smallest possible modulus: we actually have to use the laws of acceptable algebra on congruence equations to get the modulus to the smallest possible integer, and then we solve it as normal?
– The Pointer
Aug 26 at 7:16
1
1
Yes this is right, have you got the solutions for the other equations?
– Dr. Sonnhard Graubner
Aug 26 at 7:17
Yes this is right, have you got the solutions for the other equations?
– Dr. Sonnhard Graubner
Aug 26 at 7:17
Yes, I already solved them. Thanks for the clarification!
– The Pointer
Aug 26 at 7:18
Yes, I already solved them. Thanks for the clarification!
– The Pointer
Aug 26 at 7:18
1
1
Ok, nice that i could help you, have a nice day!
– Dr. Sonnhard Graubner
Aug 26 at 7:18
Ok, nice that i could help you, have a nice day!
– Dr. Sonnhard Graubner
Aug 26 at 7:18
add a comment |Â
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1
So, what is the GCD in the first example?
– Lord Shark the Unknown
Aug 26 at 7:11