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RJLiberator
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Homework Statement
For the set ℤ, define ~ as a ~ b whenever a-b is divisible by 12. You may assume that ~ is an equivalence relation and may also assume that addition and multiplication of equivalence classes is well defined where e define [a]+[ b ] = [a+b] and [a]*[ b ] = [ab] for all [a],[ b ].
Find a positive integer d such that
[d]+[5]=[0]
find a positive integer t such that
[t]+[8] =[3]
Homework Equations
3. The Attempt at a Solution [/B]These problems seem like a lot of fun. However, I'm not quite getting it.
I feel like once I understand one of these, i'll be able to understand all of the easy ones like this.
We define a ~b whenever a-b is divisible by 12.
So we are saying in the first problem d-5 has to be divisible by 12?
If d = 17 then we have 17-5 which is 12 and that is divisible by 12.
But how would [17]+[5]=[0]
In fact, how would any positive integer satisfy that? Since we have well defined addition as [a]+ = [a+b]
this would mean [d]+[5] = [d+5]
and this means [d+5] = [0], but since s must be a positive integer this could not happen...I feel like there must be something clear here that I'm missing and once I get it it will be an easily solvable problem.
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