Zongxiang Yi
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$a\equiv b\pmod{9}$ and $c+d=9$ implies $a-b\equiv 0\pmod{9}$ and $d\equiv -c \pmod{9}$ ,respectively. So $$ca+bd\equiv ca-cb\equiv c(a-b)\equiv c\cdot 0\equiv 0 \pmod{9}.$$

ans:=[n: n in [2^(2^k):k in [0..11]] | (Modexp(4,n,n^2+n+1)+Modexp(2,n,n^2+n+1)+1) mod (n^2+n+1) eq 0 ]; print #ans, ans; After running the above magma script, it outputs 12 [ 2, 4, 16, 256, 65536, ...

Theorem: $\varphi$ cannot be represented by $\sum_{d|n, d>0}f(d)$ for some funtion $f$. Proof: Assume that there exists a function $f$, such that $\varphi(n)=\sum_{d|n, d>0}f(d)$. Consider some ...

Let $b=f(a)=2+\sqrt{a-2}$, where $a\in (3,5)$. You have proved that $3<f(a)=b$. Now consider $$g(a)=a-f(a)=a-2-\sqrt{a-2}.$$ Then $g(a)$ is a increasing function since $g'(a)=1-\frac{1}{2\sqrt{a-2}}... View answer 3 votes To summary, I am going to show the following result. Main Theorem: Let$f$be the function mentioned in OP. If the peroid of$f$exists, then the peroid of$f$only has one of the two forms:$\...

Codomain is a set which the images must belong to. Range is the set which the images exactly belongs to.

Let $a \in R$. We must prove that $a$ is a zero divisor or a unit. Consider the map $\varphi_a(r)=ar$ from $R$ to $R$. One Proof If $a$ is a unit, then $\varphi_a$ is a bijection. Assume that $a$ ...

Proposition: In $\mathbb{R}^n$, if $U$ is an open set and $V$ is a closed set, then $U+V$ is an open set. Proof: Note that for any point $z\in U+V$, there exist $x\in U$ and $y \in V$ such that $z=... View answer 1 votes I think the binary operation of "mod" in the OP refers to the remainder. Let$a=q_1 p + r_1$and$b=q_2 p + r_2$, where$0\le r_1,r_2<p$. Then $$r_1=(a \bmod p),\ r_2=(b\bmod p).$$ If$\,a \bmod ...

Here is another solution which follows the direction of your professor. You can recall some thing in your another post here. We use the same notations in that post. 1) Let $n=11^2\cdot 23^2$. It is ...

Let $S(f,n)$ denote the set of solutions of congruence equation $f(x)\equiv 0 \pmod{n}$ for $f=\sum_{i=0}^{m}{a_ix^i}\in\mathbb{Z}[x]$. Denote $N(f,n)=|S(f,n)|$. Lemma 1: Let $n=\prod_{i=1}^{k}{... View answer 1 votes As the comments point out whether$2$is a primitive root of$p$depends on$p$,$a$and$y$, this answer is trying to figure out WHEN$2$is a primitive root of$p$. As $$a\equiv 2^y \pmod{p},$$ ... View answer Accepted answer 1 votes Due to the definition of events, we can treat events as sets. https://en.wikipedia.org/wiki/Event_(probability_theory) Lemma:$A-B=A-(A\cap B)$. Proof:According to the definition of the difference ... View answer Accepted answer 1 votes Lemma 1. If$f$is continuous on$[a,b]$, then for any$y_0 \in (N,M)$there exists a point$x_0 \in (a,b)$such that$f(x_0)=y_0$, where$N=\min(f(a),f(b))$and$M=\max(f(a),f(b))$. Since$f$is ... View answer 1 votes In particular, there are only two equivalence classes of$S$. One is, denoted by$\overline{0}$, $$\{ \emptyset, \{1, 2\}, \{ 1, 3\}, \{1, 4\}, \{2,3\},\{2,4\},\{3,4\},\{1,2,3,4\} \}.$$ And the other ... View answer 1 votes Use truth table. A B C A->B A->(BVC) 0 0 0 1 1 0 0 1 1 1 0 1 0 1 1 0 1 1 1 1 1 0 0 0 0 1 0 1 0 1 1 1 0 1 1 1 1 1 1 1 You can see that when A$\...

I think a simpe way is to construct two sequences $a_n$ and $b_n$, where $a_n$ are rational and $b_n$ are irrational, such that $\lim_{n\to\infty}g(a_n)=2^2$ and $\lim_{n\to\infty}g(b_n)=2^4$. Then ...

Note that $lcm(a,b,c)=lcm(lcm(a,b),c)$ and $lcm(a,b)=\frac{ab}{\gcd(a,b)}$, so what you should do the to understand the multiplication and division (https://en.wikipedia.org/wiki/Euclidean_division )...

Lemma: If $g(n)>0$, $h(n)>0$ when $n \to \infty$ and $$\lim_{n \to \infty}g(n)=a, \lim_{n \to \infty}h(n)=b,$$ then $$\lim_{n \to \infty}g(n)^{h(n)}=a^b.$$ So $$\lim_{n \to \infty}(\frac{n^2-... View answer 1 votes By Sylow's theorems, there exists a sylow 2-subgroup G_1 and a sylow 7-subgroup G_2. Note any p-subgroups G have non-trival center Z(G). Now you can choose a \in Z(G_1) and b \in Z(G_2) ... View answer 1 votes So you have x_{n+1}-x_n=c+\sqrt{x_n}-x_n. Now consider the function:$$f(x)=c+\sqrt{x}-x,x\in R.$$It follows$$f'(x)=\frac{1}{2\sqrt{x}}-1.$$You can see that when x< \frac{1}{4}, it has$$f'(...

What is "maximal" ? In general, let $(U,\le)$ be a partially ordered set. An element $m\in U$ is called maximal if $x\le m$ for all $x\in \{u\in U \mid u\le m \text{ or } m\le u\}$. In OP, $... View answer 0 votes Note that$|x|=\sqrt{x^2}$. So $$|x-5|=|3x-2|\Leftrightarrow \sqrt{(x-5)^2}=\sqrt{(3x-2)^2}.$$ Lemma:If$x,y>0$, then$x=y$if and only if$\sqrt{x}=\sqrt{y}$. Since$(x-5)^2>0$and$(3x-2)^...

Here is another method. Assume that $\lambda_i\in\mathbb{C}$, $i=1,2$ are the two igenvalues of $A$. Let $f(x)\in \mathbb{C}[x]$ be the characteristic polynomial of $A$. Then $$f(x)=\det(A-xI)=x^2-5x-... View answer 0 votes If consider functions of the form ke^x, then YES. Assume f(x)>0. As$$\mathrm{\frac{d(\log(f(x)))}{dx}}=\frac{1}{f(x)}\cdot\mathrm{\frac{df(x)}{dx}}=1,$$We have$$\int{\mathrm{\frac{d(\log(f(...

@Hagen von Eitzen shows that it holds for $m\ge 3$. Still it need to check if $$\int_1^2\frac{\sqrt{x^6+4}}{x^3}\,\mathrm dx\ge\frac54.$$ This answer will prove that it holds for $m=2$. Let $f(x)=\... View answer 0 votes For simplicity, denote$[n]=\{1,2,\cdots,n\}$. Lemma 1: Let$S$be an infinite set and$\sigma:S \to S$be a finitary permutation -- a permutation that moves only finitely many elements. Then, ... View answer 0 votes For any element$x\in \mathrm{GF}(3^3)$, there exist$a_i\in \mathbb{Z}_3$,$i=0,1,2$, such that$x=\sum_{i=0}^{2}a_i\alpha^i$. Suppose$y=\sum_{i=0}^{2}b_i\alpha^i$such that$xy=1$, where$b_i\in \...

The key point is to notice that $\lim_{n\to \infty}{a_n}=\lim_{n\to \infty}{a_{n+k}}$ for any $k\in \mathbb{Z}$. It means that the limitation does not depend on the first $k$ items. Define a sequence:...