What exactly is a mutually incongruent solution of a congruence? Can you give an example?

- anonymous

What exactly is a mutually incongruent solution of a congruence? Can you give an example?

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- ganeshie8

\(1\) and \(6\) are not mutually incongruent in mod \(5\)
because \(1\equiv 6\pmod{5}\)

- ganeshie8

\(1\) and \(3\) are mutually incongruent in mod \(5\)
because \(1\not\equiv 3\pmod{5}\)

- ganeshie8

incongruent is just an opposite of congruent

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- anonymous

ok, so two numbers m and n are mutually incongruent mod c means
m not congruent n (mod c)

- ganeshie8

Yep!

- anonymous

For this example: Find a complete set of mutually incongruent solutions of 7x≡5 (mod 11)

- anonymous

x = 7, so all solutions are x = 7 + 11k, k an integer

- ganeshie8

the answer should be just \(\{7\}\)

- anonymous

why {7} ? and not, say {18} ??

- ganeshie8

sure, you can put any single integer that is congruent to 7

- ganeshie8

but the numbers 0 to n-1 are more convenient and look better

- anonymous

But 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

there 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

every other solution is congruent to 7, so there are no other incongruent solutions

- anonymous

huhm... 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

looks good

- ganeshie8

no wait, how do you know there exists a solution ?

- anonymous

because gcd(9,15) = 3 | 12

- anonymous

I see x = 3 is one of the solution so all solution is
x = 3 + k * 15/gcd(9,15)
x = 3 + 5k

- ganeshie8

"If a solution exists", then there will be exactly 3 incongruent solutions

- ganeshie8

consider below congruence :
\[9x\equiv 2\pmod{15}\]
how many solutions ?

- anonymous

none, gcd(9,15) does not divide 2

- anonymous

right?

- ganeshie8

Yep!

- anonymous

so.....???

- anonymous

let's get back to 9x ≡ 12 (mod 15). There will be exactly 3 mutually incongruent solutions. And all solutions are x = 3 + 5k

- anonymous

what are the mutually incongruent solutions?

- ganeshie8

let k = 0,1,2

- ganeshie8

the incongruent solutions are \(\{3,8,13\}\)
they are mutually incongruent in mod \(15\)

- anonymous

how do you know to let k = 0,1,2??

- ganeshie8

they are the most convenient ones

- ganeshie8

letting k = 222, 223,224 is technically fine
but you're not keeping things simple here

- anonymous

ok, let 0 <= <= 10, then
x = 3,8,13, 18, 23, 28, 33, 38, 43, 48, 53

- ganeshie8

3 and 18 are not incongruent

- anonymous

right. I was trying to understand how to find the incongruent solutions by listing a few solutions

- anonymous

what i notice is
3 + 15 = 18
8 + 15 = 23
13 + 15 = 28

- anonymous

it's somehow like {3,8,15}, {18,23,28}

- ganeshie8

notice that any integer greater than or equal to 15 is congruent to some integer in the set \(\{0,1,2,\ldots,14\}\)

- ganeshie8

so we don't need to bother about anything greater than 14

- ganeshie8

and we don't need to bother about anything less than 0

- ganeshie8

the set of residues \(\{0, 1, 2, \ldots, n-1\}\) is our most favorite one, when somebody asks you for incongruent solutions, they expect you to give them from this set

- anonymous

So {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

mod 15 btw

- ganeshie8

{3, 8, 13} right

- anonymous

opps yeah. Typo :D

- anonymous

3 + 5k not congruent 8 + 5s not congruent 13 + 5t (mod 15), for all integer k,s,t

- ganeshie8

3 + 15k not congruent 8 + 15s not congruent 13 + 15t, for all integer k,s,t
in mod 15

- ganeshie8

notice i have changed 5k to 15k

- ganeshie8

3 + 5k refers to mod 5
3 + 15k refers to mod 15

- anonymous

I think I understand it better now although I still need to do some examples to clear up any confusion.
@ganeshie8 thank you :)

- ganeshie8

basically 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

yeah, that's what I noticed earlier when I said
3 + 15 = 18
8 + 15 = 23
13 + 15 = 28

- anonymous

so 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

you may simply say, the solutions are \(x\equiv 3\pmod{5}\)

- ganeshie8

you could also say \(x\equiv 3,8,13\pmod{15}\)

- anonymous

right right. That's exactly what I was trying to say :D

- ganeshie8

if you're not explicitly asked for the form of solutions, leaving it as \(x\equiv 3\pmod{5}\) looks neat

- anonymous

awesome. Thanks

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