4
votes
2answers
38 views

Non-negative fractions summing to $1$

Let $ d_1,\ldots, d_n \ge 2 $ be pairwise relatively prime. Are there any $ c_1,\ldots,c_n \in \mathbb{Z}_{\ge 0} $ with $ c_i \le d_i-1 $ for all $ i=1,\ldots,n $, such that $\displaystyle ...
0
votes
0answers
20 views

Adding a natural number to a normalized fraction

I am currently writing yet another rational number class where the fraction should always be normalized. When adding a natural number to a normalized fraction, it possible to get a non-normalized ...
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, ...
6
votes
1answer
122 views

Find all natural numbers such that $\sum_{k=1}^{n} \frac{n^k}{k!}$ is an integer

Find all natural numbers such that $\sum_{k=1}^{n} \frac{n^k}{k!}$ is an integer. I've tried to bring all fractions under commmon denominator and it didn't helped me much. With guessing I find out ...
1
vote
3answers
211 views

If the sum of two irreducible fractions is an integer, then the denominators are equal

I have to show the following:"If the sum of two irreducible fractions with positive denominators is an integer, then the denominators are equal." $$\frac{a}{b}+\frac{c}{d}=k, \text{ where k an integer ...
2
votes
1answer
54 views

Cyclic rearrangements of periods of certain periodic numbers

A student of mine observed the following \begin{align} \frac{1}{7}=0.\overline{142857} &\qquad \frac{2}{7}=0.\overline{285714} &\qquad \frac{3}{7}=0.\overline{428571} \\ ...
8
votes
1answer
354 views

IMO 1979 problem

The question is $$\text{If }\, p, \ q\in \mathbb{N}, \;1-\frac12+\frac13-\frac14-\dotsb-\frac{1}{1318}+\frac{1}{1319}=\frac{p}{q}.\qquad \text{Prove that } 1979\mid p.$$ So my solution went like ...
2
votes
2answers
70 views

For how many integers $a$ is $\frac{2^{10} \cdot 3 ^8 \cdot 5^6}{a^4}$ an integer?

In Mathleague $11316$ Target #$4$, the question is: For how many integers $a$ is $$\frac{2^{10} \cdot 3 ^8 \cdot 5^6}{a^4}$$ an integer?
1
vote
1answer
81 views

Any 'odd unit fraction' whose denominator is not $1$ can be represented as the sum of three different 'odd unit fractions'?

Let us call a fraction whose denominator is odd 'odd fraction'. Also, let us call an odd fraction whose numerator is 1 'odd unit fraction'. Then, here is my question. Question : Is the following ...
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 ...
0
votes
0answers
123 views

Rounding algorithm

I'm working on making some economic calculations prettier to eye, and that involves a lot of rounding, which caused me some problems. I'm aware that result of rounding depends on chosen method, but ...
1
vote
2answers
80 views

most efficient way to convert a number into a fraction

supposing I have a decimal like $$ 0.30000000000000027$$ What would be the best way to know the same number but in a fraction way like we know $\dfrac{1}{3} > 0.30 > \dfrac{1}{4}$ because ...
3
votes
1answer
485 views

Nested Division in the Ceiling Function

During class, we were introduced to a proof that used the ceiling function. We assumed (without proof) that: $$ \left\lceil{\frac{n}{2^i}}\right \rceil= ...
6
votes
1answer
338 views

Number of triples $(a,b,c)$ of positive integers such that $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{3}{4}$?

What is the number of triples $(a,b,c)$ of positive integers such that $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{3}{4}$ is: A) $16$ B) $25$ C) $31$ D) $19$ E) $34$ Note: ...
2
votes
2answers
155 views

How to write inverse of integer as sum of fractions

I was reading this article about partial fractions and at the bottom of the article there was a paragraph about integers. However, I cannot seem to get it right each time. For example: ...
4
votes
3answers
284 views

Constructing Farey sequences inductively

Objective: I'd like to prove that $F_{n+1}$ (the Farey sequence of order $n+1$) is obtained form the Farey sequence $F_n$ of order $n$ by adding all fractions of the form $\frac{a+c}{b+d}$ when ...
3
votes
1answer
93 views

How fast is a low denominator encountered, when using only mediants?

This question is (remotely) related to How to find a "simple" fraction between two other fractions?, but is not answered in that older post. Let $f_1=\frac{a}{b}$ and $f_2=\frac{c}{d}$ be ...
3
votes
1answer
95 views

Is there a direct proof of this inequality between quotients of integers?

Let $\frac{a}{b}$ and $\frac{c}{d}$ be two reduced fractions with $bc-ad > 1$ (and hence $\frac{a}{b} \lt \frac{c}{d}$) and $a,b,c,d$ positive. It is well known that there are integers $u,v$ ...
3
votes
3answers
529 views

How to find a “simple” fraction between two other fractions?

If we have two fractions $a = { a_1 \over a_2} $ and $c = {c_1 \over c_2}$ with $a<c$, how to find the fraction $b = { b_1 \over b_2 }$ , $a < b < c$ for which some measure of ...
5
votes
1answer
212 views

When is the $lcm$ of a fraction sum the actual denominator.

Consider a sum $$\frac{a}{b}+\frac{c}{d} = \frac{x}{y}$$ where each fraction is reduced. Alternatively using the familiar process of lowest common denominators, we have $$\frac{a}{b}+\frac{c}{d} = ...
2
votes
2answers
152 views

How much can a fraction reduce?

Assume $x/a$ and $y/b$ are positive fractions in it's reduced form. If $x/a+y/b=z/c$, where $z/c$ is also reduced. What can we say about $c$? Does $\frac{ab}{\gcd(a,b)^2}|c$? If it's not true. Is ...
4
votes
2answers
260 views

Bound on lcm of denominators of rational numbers that sum to 1.

This is related to the question If a finite set of rational numbers sums to one, does one of the rationals have a denominator equal to the LCM of all the denominators? Suppose $1 = ...
3
votes
1answer
270 views

If a finite set of rational numbers sums to one, does one of the rationals have a denominator equal to the LCM of all the denominators?

I was experimenting with an algorithm for generating random numbers from a discrete distribution and came across an interesting observation. Suppose that you have any finite set of rational numbers ...
8
votes
1answer
537 views

Prove that any rational can be expressed in the form $\sum\limits_{k=1}^n{\frac{1}{a_k}}$, $a_k\in\mathbb N^*$

Let $x\in\mathbb{Q}$ with $x>0$. Prove that we can find $n\in\mathbb{N}^*$ and distinct $a_1,...,a_n \in \mathbb{N}^*$ such that $$x=\sum_{k=1}^n{\frac{1}{a_k}}$$
32
votes
7answers
1k views

Bad Fraction Reduction That Actually Works

$$\frac{16}{64}=\frac{1\rlap{/}6}{\rlap{/}64}=\frac{1}{4}$$ This is certainly not a correct technique for reducing fractions to lowest terms, but it happens to work in this case, and I believe there ...
2
votes
1answer
461 views

Interesting problem on “neighbor fractions”

This is from I. M. Gelfand's Algebra book. Fractions $\displaystyle\frac{a}{b}$ and $\displaystyle\frac{c}{d}$ are called neighbor fractions if their difference $\displaystyle\frac{ad - bc}{bd}$ ...