Articles of chinese remainder theorem

Reverse of Chinese Remainder Theorem

For the following: $(102n-51) \not\equiv 2 \pmod {2,3,5,7,11,13,…,\sqrt{102n-51}}$ (That’s probably completely incorrect use of symbols, but I mean not equivalent to 2 mod any prime less than $\sqrt{102n-51}$) here are my questions: What is the first (smallest) $n$ solution? Are there infinitely many $n$ solutions? (Most importantly) Is there a way we’d know (be able […]

system of congruences proof

I’ve checked a lot of the congruency posts and haven’t seen this one yet, so I’m going to ask it. If there is a related one, I’d be happy to see it. Let $x \equiv r\pmod{m}, x \equiv s\pmod{(m+1)}$. Prove $$x \equiv r(m+1)-sm \pmod{m(m+1)}$$ So with the given conditions, we know $$x=mk_1+r$$ Then $mk_1+r \equiv […]

Chinese Remainder Theorem: solving $x^2 \equiv 1$ (mod 91).

I am trying to solve the following problem: find all solutions to the congruence $x^2 \equiv 1$ (mod 91). Already, I have solved the congruence $x^2 \equiv 1$ (mod 7) and (mod 13) and I am trying to use the Chinese Remainder Theorem however I am puzzled by how exactly to use it in this […]

Find the smallest number which leaves remainder 1, 2 and 3 when divided by 11, 51 and 91

While my preparation for exams, came across this question. “Find the smallest number which leaves remainder 1,2 and 3 when divided by 11,51 and 91” Find considerable time in solving this. I have also gone through similar questions here This site gives answer for this question as 1277 with a detailed strategy. can this be […]

systems of congruences and CRT

I want to establish an efficient method to solve linear congruences to prove the Chinese Remainder Theorem. I need a proper generalization/proof for the following ones to go to further developments. If $a$ and $b$ are integers, then $a \bmod b = {a, a-b, a+b, a-2b, a+ 2b,…}$ Let $a_1, a_2, … , a_n, b$ […]

modules with several powers $x^{y^z}$

I have an assignment(which gives no credit) which I cant solve so I would like to get some help $3^{{1001}^{1002}}\pmod {43}$ (All original numbers were replaced) Our hint was to use the Chinese-remainder-theorem but I don’t see how when $43$ is a prim number.

Solvability of $a \equiv x^2 \mod b$

Suppose you want to prove that $\exists x \in \mathbb{Z}$ with $a \equiv x^2 \mod b$. Write $b = \prod_{i = 1}^{k} p_i^{e_i}$, the prime factorisation of $b$. Why is the equivalent with finding solutions to $a \equiv x_i^2 \mod p_i$? How does one apply the Chinese Remainder Theorem here? Thanks in advance!

Sheafyness and relative chinese remainder theorem

The relative chinese remainder theorem says that for any ring $R$ with two ideals $I,J$ we have an iso $R/(I\cap J)\cong R/I\times_{R/(I+J)}R/J$. Let’s take $R=\Bbbk [x_1,\dots ,x_n]$ for $\Bbbk $ algebraically closed. If $I,J$ are radical, the standard dictionary tells us $R(I\cap J)$ is the coordinate ring of the variety $\mathbf V(I)\cup \mathbf V(J)$. Furthermore, […]

Doubts about a nested exponents modulo n (homework)

As part of my homework I am supposed to find the remainder of the division of $2^{{14}^{45231}}$ by $31$. Using the ideas explained in calculating nested exponents modulo n I tried the following: $\phi(31)=30$ since $31$ is prime. Then: $2^{{14}^{45231}}$ (mod $31$) = $2^{{14}^{45231} (\textrm{mod}30)}$ (mod $31$) Since $30 = 2 \cdot 3 \cdot 5 […]

Flaw or no flaw in MS Excel's RNG?

I have a question about my understanding of an article of B.D. McCullough (2008) about Excel’s implementation of the Wichmann-Hill random number generator (1982). First, a bit of context The Wichmann-Hill algorithm is given in AS 183 here. As one can see in the program’s comments, it’s merely three linear congruential generators (LCG) combined to […]