412.00.1.12 are relatively prime for all n.

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In summary, when a number is relatively prime for all values of n, it means that it does not share any common factors other than 1 with any value of n. This statement can be proven through mathematical proof or numerical calculation. The concept of relative primality is important in various areas of mathematics. There are many other similar statements involving relatively prime numbers, such as all prime numbers being relatively prime to each other. Finally, 412.00.1.12 cannot be relatively prime with a negative value of n, as it must be a positive integer for relative primality to hold true.
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karush
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$\tiny{412.00.1.12}$
Show that $5n+3$ and $7n+4$ are relatively prime for all n.
$$ax + by = 1$$
$\begin{array}{ll}
\textit{let} &a=5n+3 \textit{ and } b=7n+4\\
\textit{then} &(5n+3)x + (7n+4)y = 1\\
\textit{compute}&(7n+4)=(5n+3)+(2n+1)\\
&(5n+3)=2\cdot(2n+1)+(n+1)\\
&(5n+3)-(4n+2)=n+1
\end{array}$

ok no book answer but hope what I put here is sort of the idea
 
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$7(5n+3) - 5(7n+4) = 1$.
 

1. What does "412.00.1.12 are relatively prime" mean?

"412.00.1.12 are relatively prime" means that the numbers 412, 0, 1, and 12 have no common factors other than 1. In other words, they cannot be divided by any number other than 1 without leaving a remainder.

2. How do you determine if two numbers are relatively prime?

To determine if two numbers are relatively prime, you can use the Euclidean algorithm. This involves finding the greatest common divisor (GCD) of the two numbers. If the GCD is 1, then the numbers are relatively prime.

3. What is the significance of "412.00.1.12 are relatively prime" in mathematics?

The concept of relatively prime numbers is important in various mathematical fields, such as number theory and cryptography. It allows for the simplification of fractions and plays a role in determining the number of possible solutions to certain equations.

4. Can 412, 0, 1, and 12 be replaced with other numbers and still be relatively prime?

Yes, any set of numbers that have no common factors other than 1 can be considered relatively prime. For example, 5, 9, 11, and 13 are also relatively prime.

5. How is the statement "412.00.1.12 are relatively prime" relevant to real-world applications?

Relatively prime numbers are used in encryption algorithms to ensure the security of sensitive information. They are also used in creating unique identification numbers, such as credit card numbers, to prevent duplicates.

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