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Browse High School Number Theory
Stars indicate particularly interesting answers or good places to begin browsing.

Selected answers to common questions:
    Diophantine equations.
    Infinite number of primes?
    Testing for primality.
    What is 'mod'?

Divisibility of Squares of Prime Numbers [02/14/1998]
If p is a prime greater than 3, prove that p^2 leaves a remainder of 1 when divided by 12.

Divisibility of Zero Theory [10/06/1997]
A student claims that he has heard of divisibility OF zero theory... can you fill me in on this concept?

Divisibility Patterns in Pythagorean Triples [12/08/2003]
A student of mine noticed that in every Pythagorean triple, a or b is divisible by 3, a or b is divisible by 4, and a, b, or c is divisible by 5 (sometimes the same number meets more than one of the conditions). We are trying to determine if this is really true, and if so, how to prove it.

Divisibility Proof [02/16/2001]
How can I prove that if n is an odd positive integer, then 2269^n + 1779^n + 1730^n - 1776^n is an integer multiple of 2001?

Divisibility Proof [03/09/1998]
Divisibility of any given positive integer by another built from only 1's and 0's.

Divisibility Proof [10/26/1999]
How can I prove that (n^5-n) is divisible by 30, and (n^7-n) is divisible by 42, without using induction?

A Divisibility Proof; Also, a Function with Integer Values to Find [12/31/2011]
A student seeks help with two separate questions: proving that one polynomial divides another; and determining the integer values of a function given a product of its variables. Doctor Vogler invokes modular arithmetic to crack the proof, and attacks the function as a quadratic polynomial.

Divisibility Proof by Euclidean Algorithm [02/20/2003]
Let a and b be integers. Suppose that (a,b) = 1 (assuming the gcd exists). Prove that there exist integers x and y such that ax + ay = 1.

Divisibility Proof for Odd Integers [02/13/2002]
Prove that for all odd integers N, N^3 - N is divisible by 8.

Divisibility Rule for All Divisors [11/07/1999]
Is there a theorem for figuring out divisibility rules for all natural numbers?

Divisibility Tests to Find the Smallest Prime Factor of a Number [02/02/2006]
How can I quickly find the smallest positive prime divisor of 1633 without having to do lots of divisions to check the possibilities?

Divisible by 37, Comprised of Only Zeros and Ones [08/06/2013]
After solving a minimization puzzle with a brute force approach, a teacher seeks an easier method. Doctor Greenie obliges by exploiting a general divisibility rule and powers of ten.

Division by Zero: Indeterminate or Undefined? [02/23/2002]
I'm having some trouble understanding division by zero.

Division of Large Numbers [04/28/1998]
What is the remainder when 7^100 is divided by 13? Give a general strategy and an explanation.

Does Infinity Exist? [11/15/2001]
What proof do we have that infinity actually exists?

Do Rational and Irrational Numbers Alternate? [10/13/2000]
If any two non-equal real numbers "contain" an irrational, and any two non-equal real numbers "contain" a rational, do rational and irrational numbers alternate?

Double Factorial [02/22/2002]
Can you tell me what two ! marks mean in factorial questions?

Dragon's Tail [04/01/2003]
An n-dragon is a set of n consecutive positive integers. The first two-thirds of them is called the tail, the remaining one-third the head, and the sum off the numbers in the tail is equal to the sum of the numbers in the head. Find the sum of the tail of a 99,999-dragon.

Duotrigesimal (Base 32) Numbers [06/11/1999]
A unique and interesting use for base 32 or "duotrigesimal" numbers.

Egyptian Fractions [06/11/2001]
The Egyptians wrote all their fractions as a sum of different fractions with a numerator of 1. I need to find a way to work out what fractions should be added together...

Encrypting Many as One, and Decrypting Them Again [08/24/2014]
An adult seeks to encode a table of values into one number, with full recoverability. Taking a cue from random number generators, Doctor Douglas suggests a decimal representation, interleaving, and parsing protocol.

e^pi vs. pi^e [03/20/2002]
Which is greater, e^pi or pi^e? I would like to have a simple proof.

Equality Properties and What They Really Mean [07/30/2008]
In class we are shown how to square both sides of an equation or take the square root of both sides, but is there a rule like the addition property of equality that formally says those are valid steps?

Equations with a Common Root [08/22/2001]
Find all real numbers a such that the equations x^9+ax^7-(a-3)x^6-1/ 2x^2+1=0 and 2x^5+2ax^3-(2a-6)x^2+1=0 have a common root.

Equations with Rational Expressions in Two Variables [12/07/2002]
Determine all positive integers a and b that satisfy the equation: 1/a + a/b + 1/ab = 1.

Equation without a Solution [11/14/2001]
What is the solution to the equation sqrt(x) = -2 ?

Equation with Two Exponential Terms [06/27/2009]
Find all ordered pairs (a,b) for which 3^a + 7^b is a perfect square.

Equivalent Sums of Squares [07/20/2002]
Is a^2 + b^2 = c^2 + d^2 possible where a, b, c, d are positive real integers and where the pairs of squares are not identical?

Error: Division by Zero [02/12/2001]
How can I explain to my third grader that a number divided by zero is undefined? The school calculator gives the answer 0/E, and the Windows calculator gives positive infinity.

Euclidean Algorithm [10/13/1997]
Can you tell me what Euclid's theorem is in layman's terms?

Euclidean Algorithm [01/25/2003]
Given two nonzero positive integers a and b, each at most 100 digits long, use the Euclidean algorithm process to find an example of (a,b) such that they produce the longest possible chain.

Euclidean Algorithm and Linear Equations [11/03/2003]
Could you please explain step by step how to use the Euclidean Algorithm to solve a linear equation and find x and y integers?

Euclidean Algorithms [3/13/1996]
What is the Euclidean algorithm? What is a "constructible" number? What can you tell me about Diophantine equations?

Euclid's Extended Algorithm [09/16/2001]
Can you please state for me the steps of Euclid's extended algorithm in simple terms?

Euclid's Proof on the Infinitude of Primes [10/31/1995]
Which Greek mathematician proved that there is no greatest prime number?

Euler Phi Function [02/24/2002]
If p and q are prime, investigate: phi(p^n * q^m).

Euler's theorem [7/2/1996]
How do I find the inverse of a modulo m using Euler's theorem?

Even and Odd Numbers in Base 5 [02/02/2002]
How can you tell if a number in base 5 is even or odd?

Even-Digit Palindromes Divisible by 11 [12/08/1997]
Can it be proved that every even-digit palindromic number is divisible by 11?

Even - Odd Handshake Problem [05/11/2000]
How can I prove that the number of persons who have shaken an odd number of hands is even?

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