# Questions tagged [divisibility]

This tag is for questions about divisibility, that is, determining when one thing is a multiple of another thing.

4,939 questions
Filter by
Sorted by
Tagged with
22 views

### Parity of Euler's totient function

Let $S$ be a set of all numbers $k$ such that $(n, k) = 1, 1 \leq k \leq n.$ Of course, smallest element in $S$ is (by definition) $1$ and largest is $n - 1$ (since $\text{gcd}(n - 1, n) = 1$). In ...
18 views

### Polynomial Division Under Certain Remainders

Let $P(x)$ be a polynomial such that when $P(x)$ is divided by $x-17$, the remainder is $14$, and when $P(x)$ is divided by $x-13$, the remainder is $6$. What is the remainder when $P(x)$ is divided ...
38 views

### Use Fermat's Little Theorem

Find a number $0 \leq a < 73$ with $a≡9^{794}\mod 73$. I know that $a$ and $73$ are relatively prime and $a^{72}≡1 \mod73$. But I couldn't use the theorem. Can someone help me please?
16 views

### How many solutions are there to the congruence

How many solutions are there to the congruence X^4 + 5X^3 + 4X^2 - 6X - 4 ≡ 0 (mod11) with 0 ≤X ≤11? I need to find that that if there are 4 solutions or there are fewer ...
19 views

### Diophantine equations that involve Gregory coefficients: a computational exercise

In this post, for integers $k\geq 1$, we denote the Gregory coefficients as $G_k$. Wikipedia has an article for Gregory coefficients, are known as reciprocal logarithmic numbers (I add this as ...
38 views

### The length of the algorithm [closed]

The "3n + 1 algorithm" works as follows. Start with any number n. If n is even, divide it by 2. If n is odd, replace it with 3n + 1 So, for example, if we start with 5, we get the list of numbers 5,...
32 views

51 views

### Total ordering - Partially ordered set

A = {2,4,5,6,9,10,12,18,30,36,60,72} R={(a,b) | a divides b} I want to find a total order about partially ordered set(A,R). If there are multiple possible values, select a large number first. In ...
25 views

### Draw a hasse diagram about inverse R - divisibility

R = {(a,b) | a divides b} R is partially ordered set for set A. When I draw a hasse diagram about [inverse R], maybe I just change the top and bottom of the hasse diagram about [R]. Is it right?
51 views

### How to draw hasse diagram - divisibility - R and inverse R

I don't know how to draw the Hasse diagram for divisibility on the sets. A = {2,3,4,5,6,9,10} R is partially ordered set for set A. R = {(a,b) | a divides b} How to draw a hasse diagram about R and ...
47 views

52 views

### How to check if a large number is divisible by a Prime Number? [closed]

How to check if a large number is divisible by a Prime Number? Are there some divisibility rules for this?
38 views

### Finding conditions such that $4b^2 > a^2 > 3b^2$ and $b \mid (a^2-1)$ imply $b=(a+1)/2$

Consider the set of odd positive integers $a$ and $b$ such that $4b^2 > a^2 > 3b^2$ and $b \mid (a^2-1)$. Brute-force computation suggests that $a=2b-1$ is the only solution for “most” such $b$,...
35 views

### Consider set $\mathbb{Z}[\sqrt{-5}] = \{a+b\sqrt{5}i : a,b \in \mathbb{Z} \}$ show that it is a ring [duplicate]

Consider set $\mathbb{Z}[\sqrt{-5}] = \{a+b\sqrt{5}i : a,b \in \mathbb{Z} \}$. My task is to show some features listed below: Show that $\mathbb{Z}[\sqrt{-5}]$ is a ring. I would like to show that ...
28 views

### Proof of divisibility by induction [duplicate]

I've recently come across a divisibility problem that I am unable to solve. I know that most of these types of problems have fairly straightforward proof-by-induction solutions -- but for this ...