Tagged Questions

Questions on congruences, linear diophantine equations, greatest common divisor, divisibility, etc.

76 views

Show that for all integers $a,b$ and every $n>0$, $(a+b)^n ≡ a^n + b^n \pmod 2$

I want to show that, for all integers $a,b$ and every $n>0$, $(a+b)^n ≡ a^n + b^n \pmod 2$. I know that $$(a+b)^n = a^n + \dbinom{n}{1}a^{n-1}b + \cdots + \dbinom{n}{n-1}ab^{n-1} + b^n.$$ I know I ...
63 views

Does every finite cyclic group appear as a subgroup of the multiplicative group of a finite field?

Does every finite cyclic group appear as a subgroup of the multiplicative group of a finite field? In other words, given any $d \in \mathbb{N}$, can we find a prime $p$ and $k \in \mathbb{N}$ such ...
34 views

Show that λ(m)<Φ(m) for every odd composite number m.

I know that every odd composite number m factors into a set of odd primes. m=p1^(e1)p2^(e2)***pn^(en). I believe that in this situation λ(m)=[(p1-1)^(e1), (p2-1)^(e2), . . ., (pn-n)^(en)], but please ...
24 views

Verifying that $11^{λ(m)+1} ≡ 11 \pmod m$ for $m = 41 \cdot 11$

I know that $λ(m) = [40, 10] = 40$ So I need to show that $11^{41} ≡ 11 \mod 451$. From here I believe I should show that $11^{41} ≡ 11 \mod 41$ and that $11^{41} ≡ 11 \mod 11$, but I am not sure how ...
42 views

If a power of a prime number say $p^r$ does not divide $n!$ then $\frac{n}{p^r}<1$

I'm busy with a proof with $p-adic$ numbers and I need to show that if $p^r\nmid n! \implies \frac{n}{p^r} < 1$ where $p$ is prime. I need this to show $\lfloor \frac{n}{p^r} \rfloor = 0$ Any ...
139 views

Finding all possible values

we have to find all possible prime values $(p,q,r)$ such that $pq = r + 1$ $2(p^2+q^2) = r^2 + 1$ I do not know how to start looking for an answer.
475 views

Find the next divisor without remainder

I divide a value and if the remainder is not 0 I want the closest possible divisor without remainder. Example: I have: $100 \% 48 = 4$ Now I am looking for the next value which divide 100 wihtout ...
420 views

Summation involving totient function: $\sum_{d\mid n} \varphi(d)=n$ [duplicate]

Prove that:$$\sum_{d\mid n} \varphi(d)=n$$ Where $\varphi(n)$ denotes the number of positive integers $m$ less than or equal to $n$ such that $\gcd(m,n)=1$ I am lost here, any help would be ...
70 views

77 views

The number of numbers not divisible by $2,3,5,7$ or $11$ between multiples of $2310$

Looking at partitions of the natural number line of the form $P=[a,b)$, I noted that if $a$ and $b$ are multiples of $6$, there exist at least $2$ numbers in the partition which are not divisible ...
268 views

Solution of $\dfrac{a}{b}=\dfrac{a'}{b'}$ if $a,b,a',b' \in \mathbb{N}$

Let $\dfrac{a}{b}=\dfrac{a'}{b'}$ , $a,b,a',b' \in \mathbb{N}$ s.t. $a$ and $b$ have no common factors. How can we show that the only solution to this equality is $a'=na$ and $b'=nb$, $n$ is a natural ...
264 views

198 views

The Cantor Set and Triadic Expansion

let $K$ be the Cantor set. I say that a number $x$ in $[0,1]$ is triadic if $x=\frac{m}{3^n}$ for some nonnegative integers $m, n$. Let $z$ be a triadic number in $[0,1]$. Do there exist two ...
65 views

Show that the number of elements in the equivalence class is n?

Question is : Let $G$ be a finiote group of order n = $p^{\alpha}$m, where $p$ is a prime number and if $p^r$ | m but $p^{r+1} \nmid$ m Let $\mathcal M$ be the set of all subset of $G$ which have ...
121 views

Prime factorization of $10^n+1$

I was just playing with these numbers and it seems to me that the numbers of the form $10^n+1$, where $n>2$ are composite. I can prove that $10^n+1$ can't be prime unless $n$ is a power of $2$, but ...