1
vote
1answer
51 views

Calculating a Factorial Base Representation

My friend thought of a system in which each number $n$ (I will first restrict my question to positive integers $n$) is represented by a digit string $(d_l,...,d_1)$ as follows $\forall n \in ...
2
votes
1answer
32 views

Confusing sum of fractions

Question is to find the sum of: $$(\frac{1}{2^2-1})+(\frac{1}{4^2-1})+(\frac{1}{6^2-1})+(\frac{1}{20^2-1})$$ I know that $a^2-b^2=(a+b)(a-b)$, and that with this I can find the LCM to be 1995, ...
0
votes
4answers
382 views

What is the definition of a positive integer?

I am reading the book "The Number-System of Algebra (2nd edition)". At the starting of page-4 the author writes: A positive integer is a symbol for the number of things in a group of distinct ...
2
votes
3answers
207 views

If $5 \times 12 = 104$, how much is $10 \times 11$?

Question is in the title, this is for my analysis course. I don't know where to begin.
1
vote
1answer
70 views

Can we restrict the average multiplicative order of a number?

We are given a size of a number system, $s$, which is the number of components in the system. For example, the quaternions have $s=4$ components. Now, in general, we will be interested in ...
0
votes
0answers
32 views

How quickly can we find a modulated sequence of powers?

How quickly can we find an element of (at least) multiplicative order at least $p$, where $p \in \mathbb{N}$? The complete question is that we start with a number system of $s$ elements; for example ...
4
votes
1answer
574 views

How to find in which number base the operation was done by looking at the corresponding operation in decimal system?

$$23 + 25 = 51 $$ What base is used in the above addition operation ? I have 2 methods to do this Method 1 : Through equations assume base be a $$23_a + 25_a = 51_a $$ $$2a + 3 + 2a + 5 = 5a +1 ...
4
votes
3answers
293 views

Why are only fractions with denominator 2 and 5 non-repeating?

Given a rational number $\frac{n}{d}$, I understand that in the base $10$ number system, the number can be represented as a non-repeating decimal number if and only if $d$ has only prime factors of ...
-1
votes
3answers
166 views

Zero, a mystery. [duplicate]

1) Is zero even? 2) Is zero a multiple of ANY number? Ok, so I did some research on this, and; for 1) I see a whole wiki article saying that this is true, but I just can't figure out WHY? for 2) I ...
2
votes
3answers
123 views

Why does base $b$ have digits from $0$ to $b-1$?

Base system with $b \in \mathbb N$ consists of $b$ digits $d_0,d_1,d_2\dots d_{b-1}$. A number $a$ is expressed by some weighted sum of (integer) powers of $b$, where the digits $d_0,d_1,d_2\dots ...
0
votes
2answers
266 views

what is the logic behind the 'UPC -A' check digit?

In UPC-A barcode symbology, the 12th digit is known as check digit and it is used by the scanner to determine if it scanned the number correctly or not. ...
2
votes
1answer
110 views

Solving system of quadratic congruences

If you have a system ex: $ab \equiv 1 \mod 9$ $ab \equiv 3 \mod 10$ $ab \equiv 10 \mod 11$ $ab \equiv 7 \mod 12$ is there a way to determine integers $a$ and $b$?
2
votes
1answer
176 views

clarification on binary and decimal representations of integers

This is from the book: Elementary number theory, David M. Burton, page 69. Given an integer $b>1$, any positive integer $N$ can be written uniquely in terms of powers of $b$ as ...
3
votes
0answers
163 views

The Mystery of the Número Cabalístico

I recently ran into an interesting type of number that people from where I was born like to refer to as nĂșmeros cabalĂ­sticos. They are supposedly "magic" kinds of numbers that possess mystical ...
0
votes
2answers
89 views

Sum of $2$ two-digits numbers

$X$ and $Y$ are two-digit numbers. If $Y=2X+2$ and $Y=2X$ in decimal and octal system respectively, and unit digits of $X$ and $Y$ are $5$ and $2$ respectively, then how to find $X+Y$ in ...
1
vote
2answers
87 views

General approach for problem for finding sum from $1$ to $N$ when all $a$'s are replaced by $b$'s

The problem is: Find the sum of all the numbers from $1$ to $100$ when all the $6$'s are replaced by $9$'s. I need some ideas on how to approach this kind of problems? Please explain your ideas, ...