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anonymous
 one year ago
What exactly is a mutually incongruent solution of a congruence? Can you give an example?
anonymous
 one year ago
What exactly is a mutually incongruent solution of a congruence? Can you give an example?

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ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.1\(1\) and \(6\) are not mutually incongruent in mod \(5\) because \(1\equiv 6\pmod{5}\)

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.1\(1\) and \(3\) are mutually incongruent in mod \(5\) because \(1\not\equiv 3\pmod{5}\)

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.1incongruent is just an opposite of congruent

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0ok, so two numbers m and n are mutually incongruent mod c means m not congruent n (mod c)

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0For this example: Find a complete set of mutually incongruent solutions of 7x≡5 (mod 11)

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0x = 7, so all solutions are x = 7 + 11k, k an integer

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.1the answer should be just \(\{7\}\)

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0why {7} ? and not, say {18} ??

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.1sure, you can put any single integer that is congruent to 7

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.1but the numbers 0 to n1 are more convenient and look better

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0But how does the number 7 or any number congruent to 7 mod 11 capture the idea of "mutually incongruent" solution? if 7 not congruent m (mod 11), what numbers play in the role of m according to the definition of mutually incongruent?

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.1there are no other integer solutions, it is like asking for all the mutually perpendicular vectors that are solutions, if there is only one vector, you give only that.

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.1every other solution is congruent to 7, so there are no other incongruent solutions

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0huhm... ok, let me give another example where it has more than 1 mutually incongruent solution. 9x ≡ 12 (mod 15). Since gcd(9,15) = 3, there will be 3 mutually incongruent solutions

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.1no wait, how do you know there exists a solution ?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0because gcd(9,15) = 3  12

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0I see x = 3 is one of the solution so all solution is x = 3 + k * 15/gcd(9,15) x = 3 + 5k

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.1"If a solution exists", then there will be exactly 3 incongruent solutions

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.1consider below congruence : \[9x\equiv 2\pmod{15}\] how many solutions ?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0none, gcd(9,15) does not divide 2

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0let's get back to 9x ≡ 12 (mod 15). There will be exactly 3 mutually incongruent solutions. And all solutions are x = 3 + 5k

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0what are the mutually incongruent solutions?

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.1the incongruent solutions are \(\{3,8,13\}\) they are mutually incongruent in mod \(15\)

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0how do you know to let k = 0,1,2??

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.1they are the most convenient ones

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.1letting k = 222, 223,224 is technically fine but you're not keeping things simple here

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0ok, let 0 <= <= 10, then x = 3,8,13, 18, 23, 28, 33, 38, 43, 48, 53

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.13 and 18 are not incongruent

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0right. I was trying to understand how to find the incongruent solutions by listing a few solutions

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0what i notice is 3 + 15 = 18 8 + 15 = 23 13 + 15 = 28

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0it's somehow like {3,8,15}, {18,23,28}

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.1notice that any integer greater than or equal to 15 is congruent to some integer in the set \(\{0,1,2,\ldots,14\}\)

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.1so we don't need to bother about anything greater than 14

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.1and we don't need to bother about anything less than 0

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.1the set of residues \(\{0, 1, 2, \ldots, n1\}\) is our most favorite one, when somebody asks you for incongruent solutions, they expect you to give them from this set

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0So {3,8,15} are the set of mutually incongruent because 3 + 5k not congruent 8 + 5s not congruent 15 + 5t, for all integer k,s,t Is this what the incongruent solutions really mean?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.03 + 5k not congruent 8 + 5s not congruent 13 + 5t (mod 15), for all integer k,s,t

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.13 + 15k not congruent 8 + 15s not congruent 13 + 15t, for all integer k,s,t in mod 15

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.1notice i have changed 5k to 15k

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.13 + 5k refers to mod 5 3 + 15k refers to mod 15

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0I think I understand it better now although I still need to do some examples to clear up any confusion. @ganeshie8 thank you :)

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.1basically we're "lifting" the solutions from mod5 to mod15, so \(x\equiv 3\pmod{5}\) gives 3 incongruent solutions in mod 15 : \(\{3,8,13\}\)

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0yeah, that's what I noticed earlier when I said 3 + 15 = 18 8 + 15 = 23 13 + 15 = 28

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0so instead of expressing all solutions in the form of 3 + 5k, we can express all solutions in terms of mutually incongruent soltuions. I.e {x  x = 3 + 15k or x = 8 + 15k or x = 13 + 15k }

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.1you may simply say, the solutions are \(x\equiv 3\pmod{5}\)

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.1you could also say \(x\equiv 3,8,13\pmod{15}\)

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0right right. That's exactly what I was trying to say :D

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.1if you're not explicitly asked for the form of solutions, leaving it as \(x\equiv 3\pmod{5}\) looks neat
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