Articles of arithmetic

Prove that given any rational number there exists another greater than or equal to it that differs by less than $\frac 1n$

I am currently attempting to prove a claim in Hardy’s Course of Pure Mathematics and am currently stuck. I was hoping that someone would be able to provide some assistance on how to go about this. Claim: Given any rational number r and any positive integer $n$, there exists a rational number on either side […]

Significant Figures

I have learned in class that to subtract decimal numbers and keep significant figures, one just lines up the decimal, then rounds the answer according to the operand with the fewest places after the decimal. My question is how to handle an integer subtracted by a decimal. 112 – 12.0 If I add the decimal […]

Convert numbers from one base to another using repeated divisions.

I have a homework assignment for my programming class to implement an algorithm that can convert from bases 2 trough 16 to any other base from 2 trough 16 but with a few twists. What I need to understand though is how do I convert from a greater base to a lesser one (ex: 16 […]

Maximal sum of positive numbers

I’ll be grateful for any help with the foollowing question. I think the solution must be easy enough but i haven’t figured it out yet. Let a and b be positive integers such that 1) $\exists c \in \mathbb{Z}: ~~ a^2 + b^2 = c^3$ 2) $\exists d \in \mathbb{Z}: ~~ a^3 + b^3 = […]

Prove by induction that $a-b|a^n-b^n$

This question already has an answer here: Why $a^n – b^n$ is divisible by $a-b$? 9 answers

How addition and multiplication works

Lets say i am doing 12 + 13 by using the addition method that we know. i mean first we write 13 below 12, then we do 2+3 and then 1+1. The result can be validated as 25 (or true) by doing the counting manually. But for larger numbers, what is the guarantee (or proof) […]

Floor function inequality of multiplication

In a final step of a homework, I want to deduce that $$n\lfloor(n-1)!e\rfloor+2\le \lfloor n!e\rfloor+1$$ I’m unable to see whether this is true in general that $$n\lfloor a\rfloor+1\le \lfloor na\rfloor$$ where $n$ is a natural number or do I need to use power series of $e$ to do some reasoning. Thank you.

Limit of $\sqrt{x^2-6x+7}-x$ as x approaches negative infinity

What is $\lim\limits_{x\to-\infty}(\sqrt{x^2-6x+7}-x)$ ? Don’t understand how to approach this question

Factoring extremely large integers.

The question is about factoring extremely large integers but you can have a look at this question to see the context if it helps. Please note that I am not very familiar with mathematical notation so would appreciate a verbose description of equations. The Problem: The integer in question will ALWAYS be a power of […]

Finding a number given its remainder when divided by other numbers

I have this GRE question that I’d like to know how to solve. I want to solve it in as simple a way as possible, since it is GRE material. In particular, I don’t want to use “congruences” or modulo arithmetic that I came across in other posts. Here it is: When the positive integer […]