Tag: factors


Project Euler Problem 172

Posted in Mathematics

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Overview: Problem 172

How many 18-digit numbers \(n\) (without leading zeros) are there such that no digit occurs more than three times in \(n\)?

Link to Project Euler Problem 172

Background

Project Euler Problem 172 is your classic Project Euler problem: short, simple, and overwhelmingly complicated.

To nail this one, it's important to start simple - very simple. What I'll do is walk through the process of …



Tags:    computer science    mathematics    factors    sequences    euler    project euler   


Euler's Theorem, the Totient Function, and Calculating Totients By Hand

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Introduction

Today we're going to delve into a little bit of number theory.

In number theory, we are usually dealing with modular arithmetic - expressions of the form:

$$ a \equiv b \mod m $$

or

$$ f(x) \equiv 0 \mod m $$

The mod indicates we're doing modular arithmetic, which is (formally) an algebraic system called a ring, which consists of the integers 0 through m.

An analogy to modular arithmetic is the way that the sine and cosine function "wrap around," and

$$ \sin \left …


Tags:    mathematics    factors    number theory    euler   


Mad Combinatoric Castles

Posted in Mathematics

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Overview: The Problem

In an earlier post, I mentioned my efforts on Project Euler and the wide variety of problems there that can offer some profound mathematical insights.

Given that the first post covered problem #1, I thought it would be nice if the next problem cranked up the difficulty factor by an order of magnitude. Problem 502 is a very hairy combinatorics problem that required me to learn about a wide variety of combinatorial enumeration techniques.

Let's start with the problem …



Tags:    computer science    mathematics    factors    sequences    project euler