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

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In summary, the conversation discusses finding all integers a for which the polynomials (5a - 3) and (7a^2 - a + 1) have a maximum common divisor of 1. The concept of putting two functions in parentheses separated by a colon is used to represent this maximum common divisor. The Euclidean algorithm is suggested as a method for finding the GCD, and the use of the notation (a:b) = c is also mentioned.
  • #1
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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  • #2
what is the meaning of "putting the 2 functions in the parathesis separated by the colon" ??
 
  • #3
(a:b) = d
d is maximum common divisor of a and b
 
  • #4
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?
 
  • #5
How do you normally find GCDs? The Euclidean algorithm, right? Have you tried it here?
 
  • #6
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.
 

1. What is a maximum common divisor (MCD)?

A maximum common divisor, also known as a greatest common divisor (GCD), is the largest positive integer that divides evenly into two or more numbers.

2. How is the MCD calculated?

The MCD can be calculated using the Euclidean algorithm, which involves repeatedly dividing the larger number by the smaller number until the remainder is 0. The last non-zero remainder is the MCD.

3. What is the difference between MCD and least common multiple (LCM)?

The MCD is the largest number that divides evenly into two or more numbers, while the LCM is the smallest number that is a multiple of two or more numbers. In other words, the MCD is a divisor, while the LCM is a multiple.

4. Can the MCD be negative?

No, the MCD is always a positive number. If one of the numbers being divided is negative, the MCD will still be positive.

5. How is the MCD used in mathematics?

The MCD is used in various mathematical concepts such as simplifying fractions, finding equivalent fractions, and solving equations with multiple variables. It is also used in real-life applications such as simplifying recipes and calculating the lowest common denominator in fractions.

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