Subring Math Problem: Find Number of Elements in \mathbb{Z}_{2000} Subring

[roam]

Homework Statement

Let R be a ring and a be an element of R. Let $$S= \left\{ x \in R: ax=0_R \right\}$$. S is a subring of R.

Let $$R= \mathbb{Z}_{2000}$$ and $$a=850$$. Determine the elements of the subring S as defined previously. How many elements are in S?

The Attempt at a Solution

The elements of the subring S will be elements x from $$\mathbb{Z}_{2000}$$ such that $$850.x=0_R$$.

And I think since 850x=0-5000n, $$x= \frac{2000}{850} n = \frac{40}{21} n$$ then

n=k.21

But what I do I need to do to find the number of elements in S? Is there a quick way of finding this?

 

[fzero]

What's the largest multiple of 21 in Z/2000? What k does this correspond to? The other elements of S follow from this.

 

[roam]

What's the largest multiple of 21 in Z/2000? What k does this correspond to? The other elements of S follow from this.

The largest multiple of 21 $$\mathbb{Z}_{2000}$$ is 1995. It ocrresponds to k=95, since 21x95=1995. What do you mean "other elements of S follow from this"? How do I need to figure out how many elements are in S?

 

[fzero]

The largest multiple of 21 $$\mathbb{Z}_{2000}$$ is 1995. It ocrresponds to k=95, since 21x95=1995. What do you mean "other elements of S follow from this"? How do I need to figure out how many elements are in S?

Sorry, I thought your method actually determined the elements of S. I would look at the prime factorizations 2000 = 2⁴ 5³, 850 = 2 5² 17. Now, by comparing the prime factorizations, what is the smallest x (call it x_g) such that a x_g = 0_R? Now note that all multiples m x_g are also in S.

 

[roam]

Sorry, I thought your method actually determined the elements of S. I would look at the prime factorizations 2000 = 2⁴ 5³, 850 = 2 5² 17. Now, by comparing the prime factorizations, what is the smallest x (call it x_g) such that a x_g = 0_R? Now note that all multiples m x_g are also in S.

Firstly, how do you determine this x_g from the prime factorization? Also, how does it help to determine the number of elements in S?

 

[fzero]

Firstly, how do you determine this x_g from the prime factorization? Also, how does it help to determine the number of elements in S?

The condition $$a x =0_R$$ requires that $$a x = 2^4 5^3 k$$ for some k. Looking at the prime factors in a leads us to conclude that $$x \in S$$ have the form $$x_{k'}=2^m 5^n k'$$. $$m,n$$ are easily determined, while the $$k'=1,\ldots k'_{\text{max}}$$ are constrained by the condition that $$x\in R$$.

 

[roam]

The condition $$a x =0_R$$ requires that $$a x = 2^4 5^3 k$$ for some k. Looking at the prime factors in a leads us to conclude that $$x \in S$$ have the form $$x_{k'}=2^m 5^n k'$$. $$m,n$$ are easily determined, while the $$k'=1,\ldots k'_{\text{max}}$$ are constrained by the condition that $$x\in R$$.

Could you please explain a bit more and maybe give some examples? Because I'm very confused... I think all of the elements which will be zero must be factors of 2000=2.5².17. I mean 850 x (something x k)=2000k' Do I need to try these:

400x 2⁴.5 x k
200 x 2³.5² k
1000 x 2⁴.5³ k

for different k's.

 

[fzero]

Since $$850 = (2) (5^2) ( 17)$$, we compute

$$a x_{k'} = 2^{m+1} 5^{n+2} 17 k' .$$

This is $$0 (\mod 2000)$$ if $$m=3,n=1$$, so

$$x_{k'} = 40 k'.$$

Note that (850)(40)=(17)(2000), so your intuition is correct. Since 17 is prime, 40 is the smallest integer for which this works. Now $$x_{50} = 2000 = 0_R$$ gives us $$k'_{\text{max}}$$.