See also the
Dr. Math FAQ:
0.9999 = 1
0 to 0 power
n to 0 power
0! = 1
dividing by 0
Browse High School Number Theory
Stars indicate particularly interesting answers or
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Selected answers to common questions:
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
- 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 =
- 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
- 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
- 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
- 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
- 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?