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 […]

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 […]

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 […]

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 = […]

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

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) […]

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.

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

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 […]

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 […]

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