What are the possible values of a for which (5a - 3:7a^2 - a + 1) = 1?

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The discussion centers on finding integer values of 'a' for which the expression (5a - 3 : 7a^2 - a + 1) equals 1, indicating that the two polynomials are relatively prime. Participants clarify that the notation (a:b) refers to the greatest common divisor (GCD) of the two expressions, and if their GCD is 1, they are co-prime. There is confusion regarding the introduction of 'd' and how to apply the Euclidean algorithm to determine the GCD in this context. The conversation emphasizes the need for clarity in mathematical notation and definitions. Ultimately, the goal is to identify integer solutions for 'a' that satisfy the given condition.
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Find all a(integers) that (5a - 3:7a^2 - a + 1) = 1

I only know that

d|7a^2 - a + 1
d|5a - 3
 
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what is the meaning of "putting the 2 functions in the parathesis separated by the colon" ??
 
(a:b) = d
d is maximum common divisor of a and b
 
Correct me if I'm wrong, but aren't you asking for which values of a are the polynomials (5a - 3) and (7a^2 - a + 1) relatively prime? (5a - 3 : 7a^2 - a + 1) = 1 is what you have written. That doesn't quite mesh with your thread title so I'm a little lost. For example, where did "d" come from?
 
How do you normally find GCDs? The Euclidean algorithm, right? Have you tried it here?
 
If (a:b) = 1 then a and b are co-prime
Also (a:b) = c then c|a and c|b
and exists d so that d|a , d|b and d|c
thats the d I try to use.
 

Thread 'Finding the number of ways to arrange identical balls in a circle (3 different colors)'

I tried to combine those 2 formulas but it didn't work. I tried using another case where there are 2 red balls and 2 blue balls only so when combining the formula I got ##\frac{(4-1)!}{2!2!}=\frac{3}{2}## which does not make sense. Is there any formula to calculate cyclic permutation of identical objects or I have to do it by listing all the possibilities? Thanks
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Thread 'Greatest possible value of a constant in polynomial'

Since ##px^9+q## is the factor, then ##x^9=\frac{-q}{p}## will be one of the roots. Let ##f(x)=27x^{18}+bx^9+70##, then: $$27\left(\frac{-q}{p}\right)^2+b\left(\frac{-q}{p}\right)+70=0$$ $$b=27 \frac{q}{p}+70 \frac{p}{q}$$ $$b=\frac{27q^2+70p^2}{pq}$$ From this expression, it looks like there is no greatest value of ##b## because increasing the value of ##p## and ##q## will also increase the value of ##b##. How to find the greatest value of ##b##? Thanks
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