The discussion revolves around solving the congruence equation 42x ≡ 12 (mod 90), which falls under the subject area of abstract algebra. Participants explore the implications of the equation and the nature of the solutions, particularly focusing on whether fractional answers are permissible.
The discussion has progressed through various interpretations and approaches to the problem. Some participants have provided guidance on how to handle the modular arithmetic involved, while others have raised questions about the existence of inverses and the implications of the calculations performed. There is an ongoing exploration of the relationship between the concepts of abstract algebra and discrete mathematics.
Participants note that the expected result should be an integer between 0 and 89, and they discuss the constraints of working with modular arithmetic, particularly regarding the existence of inverses in certain cases.
I guess the result is expected to be an integer between ##0## and ##89##. What does ##42x \equiv 12 \mod 90## mean as an equation?Math100 said:Homework Statement:: Solve the congruence 42x \equiv 12 (mod 90).
Relevant Equations:: None.
42x-90x=-48x
x \equiv 12/-48 (mod 90)
Simplify
x \equiv -1/4 (mod 90)
Is this the correct answer for the given abstract algebra problem?
No. Your fraction is ##-(1/4)=(-1)\cdot 4^{-1}=89 \cdot 4^{-1}## and you have to calculate the inverse of ##4## modulo ##90##. However, ##4## has no inverse modulo ##90##, i.e. your result doesn't even exist.Math100 said:So a fraction in answer isn't allowed in this type of problem?
Now we need ##7^{-1} \mod 15##, i.e. which number multiplied by ##7## leaves the remainder ##1## modulo ##15\,?##Math100 said:I got 7x=15n+2. And how do I solve it from here?
That would be ##-1##. But it will do. What do we get from multiplication by ##2## on both sides?Math100 said:2, because 2 multiplied by 7 is 14.
And this means?Math100 said:14x=30 n+4
Check it out: What is ##42\cdot 11 ## divided by ##90##? Which remainder do you get?Math100 said:Is that, above, the correct answer?
... which is what we were looking for.Math100 said:462/90=5.13333333...
5×90=450
462-450=12
So the remainder is 12.
If you go through this thread, then you'll see that we only used basic arithmetic. Originally we had ##42x\equiv 12\;(90)## and the first impulse would be to ask for the inverse of ##42## in order to divide the equation. But there is no such inverse as ##42## and ##90## aren't coprime. But division by ##6## yields the same problem: ##7x\equiv 2 \;(15)## and now ##7## and ##15## are coprime. This means there is an inverse of ##7##. You can use Euclid's algorithm to find this inverse, which you can find e.g. on Wikipedia (Bezout's Lemma). But for small numbers like ##15## it is easier to just check them. I mean it is not too difficult to see that ##7\cdot 13 =91 =6\cdot 15 +1## so that ##7^{-1} \equiv 13\;(15)##. Now we can divide to separate ##x##:Math100 said:Okay. Now we just solved the problem. But is there an actual, formal formula for this type of abstract algebra problem?
Yes. I mean: both. Maybe a bit more discrete mathematics as it tends toward applied mathematics. Such calculations are used e.g. in the theory of error correcting codes. Abstract algebra emphasizes more the theoretical part: Why is modulo ##90## only a ring, whereas modulo ##15## is a field? The difference is: we can divide all numbers (except zero) in a field, but not necessarily in a ring.Math100 said:Okay. One more question, is this an example of abstract algebra problem or discrete mathematics problem?
This depends on your goals. I hope my answer got you an impression of the difference between the two.I'm considering to take an upper division mathematics course at a four year institution this Fall but don't know if Abstract Algebra will be the next course after Ordinary Differential Equations.