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.
The equation for Congruence Solution is 42x-90x=-48x Simplified.
The simplified form of the equation is -48x.
There is one variable, x, in the equation.
The solution for x is any real number, as long as it satisfies the equation -48x=0.
Congruence Solution refers to the process of finding a value for the variable that makes the equation true.