This must be a basic question. But i need some help. What is the number of decimal places that needs to be considered normally in division operations in order to represent the dividend value as a multiple of divisor and quotient by rounding off. Example: Say i want to divide 3475934 and 3475935 by 65536. […]

Find all the numbers $n$ such that $\frac{12n-6}{10n-3}$ can’t be reduced. Attempt: It can’t be reduced when $\gcd(12n-6,10n-3)=1$ Here $(a,b)$ denotes $\gcd(a,b)$ $$(12n-6,10n-3)=(12n-6,2n-3)=(12n-6,12n-18)=(12n-6,12)$$ $\Longrightarrow$ It can’t be redused when $12\nmid 12n-6$ i.e when $12n\not\equiv 6\pmod{12}$ Theorem: for $ax\equiv b \pmod n$ there is a solution iff $d\mid b$ where $d=\gcd(a,n)$. In this case $\gcd(12,12)=12$ so […]

This question already has an answer here: bijection between $\mathbb{N}$ and $\mathbb{N}\times\mathbb{N}$ [duplicate] 3 answers

It is easy to show that every natural number $n$ can be written as $n = \frac{xy}{x+y}$ with $x,y\in \mathbb{N}$ by setting $x = y = 2n$. Now I experimented a little bit with numbers of the form $n=\frac{xyz}{x+y+z}$ and it seems that every natural number $n$ can be written this way with $x,y,z \in […]

I know that I need to reduce $7^{12341} \pmod {1000}$ By Euler I have $7^{\phi(1000)}\equiv 7^{400}\equiv1\pmod{1000}$ That leaves me with the monster $7^{341}\pmod{1000}$ Is there a way to reduce this smoothly without working $7^2, 7^4, 7^8$ etc manually ?

I’m working on some Chinese Remainder problems and it doesn’t seem like I have the procedure down correctly. I’ll list the steps I’m taking so hopefully someone can spot what it is I’m doing wrong. Find the least nonnegative solution of each system of congruences below. $x \equiv 3 \space mod \space 4$ $x \equiv […]

I made a proof by contradiction. Suppose $δ=(a+b,\operatorname{lcm}[a,b])$ and let it be that $δ\neq(a,b)$. Then $\exists ε\big(ε=(a,b) \land ε\gt δ \big) \implies ε|a \land ε|b \implies ε|(a+b)$. It is also true that $ε|\operatorname{lcm}[a,b]$. By the two previous statements, we get that $ε|(a+b,\operatorname{lcm}[a,b])\implies ε|δ$. This is absurd since $ε>δ$. Thus $δ=(a,b)$. Is it correct? I wonder […]

This problem is taken from Ivan Niven’s “An Introduction to the Theory Of Numbers”. Show that ${{p^\alpha-1}\choose{k}} \equiv ({-1})^k\pmod p$. Note: This is not similar to this one, as $k! | p^\alpha$ is possible. My Attempt: We proceed by induction on k: Let, ${{p^\alpha-1}\choose{k-1}}=r$.Let $k=p^t*q \ : t<\alpha$. $(k,p^\alpha-k)=p^t$. $${{p^\alpha-1}\choose{k}} = {{p^\alpha-1}\choose{k-1}}*\frac{p^\alpha-k}{k}={{p^\alpha-1}\choose{k-1}}*\frac{p^{\alpha-t}-q}{q}$$. So, by induction […]

Find $v_p\left(\binom{ap}{bp}-\binom{a}{b}\right)$, where $p>a>b>1$ and $p$ odd prime. Here $v_p(k)$ denotes the largest $\alpha\in\mathbb Z_{\ge 0}$ s.t. $p^\alpha\mid k$. We have $p\nmid\binom{ap}{bp}$ and $p\nmid \binom{a}{b}$. $$\binom{ap}{bp}-\binom{a}{b}=\frac{(ap)!b!(a-b)!-a!(bp)!(ap-bp)!}{(bp)!(ap-bp)!b!(a-b)!}$$

Let $c$ be a $3^n$ digit number whose digits are all equal. Show that $3^n$ divides $c$. I have no idea how to solve these types of problems. Can anybody help me please?

Intereting Posts

Residue at infinity (complex analysis)
Method of Steepest Descent and Lagrange
Exciting games and material to motivate children to math
Distribution of prime numbers. Can one find all prime numbers?
Beautiful cyclic inequality
Evaluating $\int_{0}^{1} \sqrt{1+x^2} \text{ dx}$
Notation for n-ary exponentiation
Fourier series is to Fourier transform what Laurent series is to …?
When do we have $Rad(I)=I$ for an ideal $I$ of a ring $R$?
Unconventional mathematics books
Olympiad Inequality Problem
Are surreal numbers actually well-defined in ZFC?
Does Hartshorne *really* not define things like the composition or restriction of morphisms of schemes?
If $(x_n) \to x$ then $(\sqrt{x_1x_2\cdots x_n}) \to x$
Deriving cost function using MLE :Why use log function?