. Suppose gcd(a, m) = d and that m > 1. Consider the congruence ax ≡ b (mod m). Should there be a...
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. Suppose gcd(a, m) = d and that m > 1. Consider the congruence ax ≡ b (mod m). Should there be a solution for every choice of b?
On sums involving Euler's totient functionWhy $0$ in number $50$ is not a significant digit?Conjecture about primes and the factorial: for all primes $p>5$, must there exist a prime $q<p$ such that $qequiv m!pmod p$ for some $2<m<p$?What is the remainder when $a$ is divided by 6?Is there an elementary proof that $y^2=8x^4+1$ has no integral solution for $xge2$?Linear combinations of two numbers can only be multiples of their gcd?Solutions of $varphileft(n^nsigma(n)right)=varphi(n^{n+1})$, where $sigma(n)$ is the sum of divisors and $varphi(n)$ the Euler's totientOn variations of Erdős squarefree conjecture: presentation and a question as a simple caseCongruence of a power with 1, $a^x equiv 1 mod (p^alpha)$What mathematical consequences might there be if Euler Mascheroni constant is rational?
$begingroup$
If yes, prove your claim; if not, give a counter example
X is not told so I assume it can be any orbitrary number. b is also abitrary, with that being said, isn't true due to those 2 factors?
modular-arithmetic totient-function eulers-constant
asked yesterday
Kayy WangKayy Wang
314
$endgroup$
add a comment |
$begingroup$
If yes, prove your claim; if not, give a counter example
X is not told so I assume it can be any orbitrary number. b is also abitrary, with that being said, isn't true due to those 2 factors?
modular-arithmetic totient-function eulers-constant
asked yesterday
Kayy WangKayy Wang
314
$endgroup$
$begingroup$
$x$ is an unknown. $axequiv b$ is an equation, which may or may not have solutions. What the question is asking is whether the exact value of $b$ affects whether the equation has solutions..
$endgroup$
– Arthur
yesterday
$begingroup$
1) It's always a good idea to put the question in the text, not just in the title. 2) Are you sure that $m>1$ is given, $d>1$ would make much more sense here?
$endgroup$
– Ingix
yesterday
add a comment |
$begingroup$
If yes, prove your claim; if not, give a counter example
X is not told so I assume it can be any orbitrary number. b is also abitrary, with that being said, isn't true due to those 2 factors?
modular-arithmetic totient-function eulers-constant
asked yesterday
Kayy WangKayy Wang
314
$endgroup$
If yes, prove your claim; if not, give a counter example
X is not told so I assume it can be any orbitrary number. b is also abitrary, with that being said, isn't true due to those 2 factors?
modular-arithmetic totient-function eulers-constant
modular-arithmetic totient-function eulers-constant
asked yesterday
Kayy WangKayy Wang
314
asked yesterday
Kayy WangKayy Wang
314
asked yesterday
Kayy WangKayy Wang
314
asked yesterday
Kayy WangKayy Wang
314
asked yesterday
Kayy WangKayy Wang
314
314
$begingroup$
$x$ is an unknown. $axequiv b$ is an equation, which may or may not have solutions. What the question is asking is whether the exact value of $b$ affects whether the equation has solutions..
$endgroup$
– Arthur
yesterday
$begingroup$
1) It's always a good idea to put the question in the text, not just in the title. 2) Are you sure that $m>1$ is given, $d>1$ would make much more sense here?
$endgroup$
– Ingix
yesterday
add a comment |
$begingroup$
$x$ is an unknown. $axequiv b$ is an equation, which may or may not have solutions. What the question is asking is whether the exact value of $b$ affects whether the equation has solutions..
$endgroup$
– Arthur
yesterday
$begingroup$
1) It's always a good idea to put the question in the text, not just in the title. 2) Are you sure that $m>1$ is given, $d>1$ would make much more sense here?
$endgroup$
– Ingix
yesterday
$begingroup$
$x$ is an unknown. $axequiv b$ is an equation, which may or may not have solutions. What the question is asking is whether the exact value of $b$ affects whether the equation has solutions..
$endgroup$
– Arthur
yesterday
$begingroup$
$x$ is an unknown. $axequiv b$ is an equation, which may or may not have solutions. What the question is asking is whether the exact value of $b$ affects whether the equation has solutions..
$endgroup$
– Arthur
yesterday
$begingroup$
1) It's always a good idea to put the question in the text, not just in the title. 2) Are you sure that $m>1$ is given, $d>1$ would make much more sense here?
$endgroup$
– Ingix
yesterday
$begingroup$
1) It's always a good idea to put the question in the text, not just in the title. 2) Are you sure that $m>1$ is given, $d>1$ would make much more sense here?
$endgroup$
– Ingix
yesterday
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
No, it has not. For instance, the congruence $;8xequiv 5pmod{12}$ has no solution, because it would imply that $5$ is divisible by $4$.
What can be said is this:
The congruence $;axequiv bpmod m$ has a solution if and only if $bequiv 0mod d=gcd(a,m)$.
In this case, the congruence is equivalent to
$$frac ad, xequivfrac bdmodfrac md. $$
answered yesterday
BernardBernard
122k741116
$endgroup$
add a comment |
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1 Answer
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1 Answer
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$begingroup$
No, it has not. For instance, the congruence $;8xequiv 5pmod{12}$ has no solution, because it would imply that $5$ is divisible by $4$.
What can be said is this:
The congruence $;axequiv bpmod m$ has a solution if and only if $bequiv 0mod d=gcd(a,m)$.
In this case, the congruence is equivalent to
$$frac ad, xequivfrac bdmodfrac md. $$
answered yesterday
BernardBernard
122k741116
$endgroup$
add a comment |
$begingroup$
No, it has not. For instance, the congruence $;8xequiv 5pmod{12}$ has no solution, because it would imply that $5$ is divisible by $4$.
What can be said is this:
The congruence $;axequiv bpmod m$ has a solution if and only if $bequiv 0mod d=gcd(a,m)$.
In this case, the congruence is equivalent to
$$frac ad, xequivfrac bdmodfrac md. $$
answered yesterday
BernardBernard
122k741116
$endgroup$
add a comment |
$begingroup$
No, it has not. For instance, the congruence $;8xequiv 5pmod{12}$ has no solution, because it would imply that $5$ is divisible by $4$.
What can be said is this:
The congruence $;axequiv bpmod m$ has a solution if and only if $bequiv 0mod d=gcd(a,m)$.
In this case, the congruence is equivalent to
$$frac ad, xequivfrac bdmodfrac md. $$
answered yesterday
BernardBernard
122k741116
$endgroup$
No, it has not. For instance, the congruence $;8xequiv 5pmod{12}$ has no solution, because it would imply that $5$ is divisible by $4$.
What can be said is this:
The congruence $;axequiv bpmod m$ has a solution if and only if $bequiv 0mod d=gcd(a,m)$.
In this case, the congruence is equivalent to
$$frac ad, xequivfrac bdmodfrac md. $$
answered yesterday
BernardBernard
122k741116
answered yesterday
BernardBernard
122k741116
answered yesterday
BernardBernard
122k741116
answered yesterday
BernardBernard
122k741116
122k741116
add a comment |
add a comment |
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$begingroup$
$x$ is an unknown. $axequiv b$ is an equation, which may or may not have solutions. What the question is asking is whether the exact value of $b$ affects whether the equation has solutions..
$endgroup$
– Arthur
yesterday
$begingroup$
1) It's always a good idea to put the question in the text, not just in the title. 2) Are you sure that $m>1$ is given, $d>1$ would make much more sense here?
$endgroup$
– Ingix
yesterday