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

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Homework Help Overview

The problem involves determining the integer values of \( a \) for which the expression \( (5a - 3 : 7a^2 - a + 1) = 1 \) holds true, indicating a relationship between two polynomials.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants discuss the meaning of the colon in the expression, with some suggesting it refers to the greatest common divisor (GCD) of the two polynomials. Others question the clarity of the original poster's intent and the introduction of the variable \( d \).

Discussion Status

The discussion is exploring the definitions and implications of the expression involving the polynomials. Some participants have offered insights into the concept of co-primality and the use of the Euclidean algorithm for finding GCDs, while others seek clarification on the original question and its formulation.

Contextual Notes

There appears to be some confusion regarding the notation used in the problem, as well as the specific values of \( a \) being sought. The discussion reflects varying interpretations of the mathematical relationships involved.

kezman
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Find all a(integers) that [tex](5a - 3:7a^2 - a + 1) = 1[/tex]

I only know that

[tex]d|7a^2 - a + 1[/tex]
[tex]d|5a - 3[/tex]
 
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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.
 

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