If n = x(mod 10x+1) n,x>0

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In summary, the conversation discusses the possibility of finding a direct relation between n and x in terms of some parameter in order to solve the congruence. The person mentions having 9 more similar equations and the potential for factorizing any number if the congruence can be solved. They ask if modern mathematics is needed to solve the congruence, and DonAntonio suggests a potential relation between n and x but notes that there is no further direct relation beyond what is shown.
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sparsh12
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--> if n = x(mod 10x+1) n,x>0 (where = is symbol used in "congruence" not equality)

then, is there a way to find some direct relation between n and x, in terms of some parameter?

--> I have 9 more such equations and if i am able to solve either one i would be able to solve all of them.

--> And this woud help me factorise any number without searching. But only if above congruence could be solved.

what am I supposed to do to solve the congruence? Some modern mathematics?
 
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sparsh12 said:
--> if n = x(mod 10x+1) n,x>0 (where = is symbol used in "congruence" not equality)

then, is there a way to find some direct relation between n and x, in terms of some parameter?

--> I have 9 more such equations and if i am able to solve either one i would be able to solve all of them.

--> And this woud help me factorise any number without searching. But only if above congruence could be solved.

what am I supposed to do to solve the congruence? Some modern mathematics?


[itex]n=x\pmod{10x+1}\Longrightarrow n=x+k(10x+1)\,,\,\,k\in Z\Longrightarrow n=x(10k+1)+ k[/itex]...can you see any "direct" relation between n and x? Because beyond what the last equation shows, I can't.

DonAntonio
 

1. What does the notation "n = x(mod 10x+1)" mean?

The notation "n = x(mod 10x+1)" means that the number n is congruent to the number x modulo (or with respect to) the expression 10x+1. In other words, when n is divided by 10x+1, the remainder is equal to x.

2. Can n and x be any positive numbers?

Yes, n and x can be any positive numbers as long as they satisfy the equation n = x(mod 10x+1). This notation is often used in number theory and modular arithmetic to represent congruences between numbers.

3. How can this equation be solved for n?

To solve for n, we can use the division algorithm to express n as n = 10kx + k + x, where k is an integer. This means that n is equal to x plus some multiple of 10x+1. Therefore, any value of n that satisfies this equation is congruent to x modulo 10x+1.

4. What is the significance of using 10x+1 in the expression?

The expression 10x+1 is often used in modular arithmetic because it is relatively prime (has no common factors) with 10. This means that it can help generate a complete residue system (a set of numbers that leave different remainders when divided by 10) which is useful in solving problems involving congruences.

5. Can this equation be generalized to include more variables?

Yes, this equation can be generalized to include more variables, such as n = x(mod ax+b), where a and b are integers. This notation is often used in modular arithmetic to represent congruences between numbers with multiple variables.

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