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'?
- Cubes as Differences of Squares [07/04/2002]
Prove that the cube of any positive integer is equal to the
difference of the squares of two integers.
- Cute Numbers [08/29/2003]
I have found two definitions of a cute number. Which is correct?
- Cyclic Redundancy Check [06/26/2002]
I understand how cyclic redundancy checks work, but I fail to see how
appending zeros to the message string (before the division) provides an
- Dates that Read the Same Backwards and Forwards [02/02/2010]
A student sees a palindrome in the date 01 02 2010, and wonders how to
generate all such palindromic dates. Building on another math doctor's
work with date arithmetic, Doctor Carter shares a program written in C,
then goes on to explain the purpose of each line of code.
- Decimal Expansion of a Reciprocal [10/23/2001]
1/(X + Y + Z) = 0.XYZ.
- Dedekind Cuts [10/23/1996]
What is a Dedekind cut?
- Defining 0/0 [01/29/2001]
I convinced my teacher that 0/0 must be defined, since our math laws say
that anything divided by itself equals 1. Shouldn't 0/0 = 1?
- Definition of Floating Point Data [07/02/2001]
What are 'floating point data'? How do they differ from an integer? What
are some examples?
- Definitions of Advanced Concepts [11/13/1998]
Can you give me definitions for: Pythagorean Triplets, Principle of
Duality, Euclid's Elements, Cycloid, Fermat's Last Theorem?
- Definitions: Relatively Prime, Proper Factor [9/11/1996]
What does it mean to be relatively prime? What is a proper factor?
- Density Property of Rational Numbers [09/21/2001]
How is the density property of rational numbers proven?
- Deriving Properties of Fractions [08/10/2003]
Derive the rule for multiplying fractions, that a/b x c/d = ac/bd,
using lower-level properties of multiplication and rational numbers.
- Determining Factors of a 3998-digit Number [08/11/1999]
Let N = 111...1222...2, where there are 1999 digits of 1 followed by 1999
digits of 2. How can I express N as the product of four integers, each of
which is greater than 1?
- Determining If a Large Number is Divisible by 11 [10/22/2003]
I just learned a trick to decide whether a large number is divisible by
11 or not. Why does the trick work?
- Determining Primes by Their Square Roots [06/13/2001]
My problem has to do with determining if a very large number is a prime.
- Diagonal Sum in Pascal's Triangle [04/02/2001]
Find the sum of the reciprocals of the diagonals in Pascal's triangle.
- Difference of Square Numbers [07/18/2008]
Can one number ever be represented as two distinct differences of
squares? Or is every difference of square numbers unique?
- Different Infinities [02/19/1997]
How many different infinities are there?
- Digital Computers and Binary [07/02/2000]
How do digital computers use the binary number system?
- Digit Patterns of the Powers of 5 [09/14/1998]
Why is there a pattern in the last digits of the powers of 5?
- Digits of a Square [05/26/2001]
If the tens digit of a^2 (a is an integer) is 7, what is the units
- Digits Sums, Mod Proofs, Olympiad Squares, and Equal Roots [12/15/2011]
A student seeks help with four different number theory proofs. Doctors Carter and
Vogler offer observations, textbook recommendations, and other guidance.
- Diophantine by Process of Elimination [01/10/2015]
A teen knows the integer solutions to 2^x - 3^y = 5, but seeks proof. Doctor Vogler
steps through an approach that starts by reducing the equation mod m, where neither
of the bases 2 and 3 has an order.
- Diophantine Equations [11/17/1997]
We have searched the Web for information about Diophantine equations.
- Diophantine Equations [06/29/2001]
Find rational x and y such that x^2+x^2*y^2 and y^2+x^2*y^2 are perfect
squares, or, more simply, x^2+x^2*y^2 = m^2 and y^2+x^2*y^2 = n^2, where
n and m are rational numbers.
- Diophantine equations in Number Theory [01/24/2001]
If a and b are relatively prime positive integers, prove that the
Diophantine equation ax-by = c has infinitely many solutions in the
- Diophantine Equations in Three Variables [10/30/2004]
I need to know how to get positive integer solutions of two
Diophantine equations having three variables. For example: 2x + 3y +
7z = 32 ; 3x + 4y - z = 19. (Give the positive set of triples for the
- Diophantine Equations, Step by Step [10/01/2002]
Find all positive integer solutions to 43x + 7y + 17z = 400.
- Diophantine Equation to Find Perfect Square Values [03/19/2008]
Given a long polynomial such as 4x^4 + x^3 + 2x^2 + x + 1, how can I
find positive integers that would produce a perfect square value when
substituted into the polynomial?
- Direct Conversion of Binary to Octal [05/14/2002]
How can you convert from base 2 to base 8 without going through base
- Discrete Logarithm Problem [10/13/2004]
Given a === b^c mod N. When a, b, and N are given, can we find c?
- Displaying Large Repetends on Small Calculators [05/31/2002]
How can I find a 16-digit repetend using an 8-digit calculator?
- Distance between Points on a Line [10/02/2002]
When the 10 distances between 5 pairs of points on a line are listed
from smallest to largest, the list reads: 2,4,5,7,8,k,13,15,17,19.
What is the value of k?
- Dividing 29/49 [08/30/1997]
Can I divide 29/49 out until it repeats itself or terminates without
using long division?
- Divisibility by 11: Proof [02/12/2002]
Prove that a positive integer n is divisible by 11 if and only if the
alternating sum of its digits is divisible by 11.
- Divisibility by 37 [11/08/1997]
Take a 3-digit number and add to that its "rotation". Prove that the sum
can always be divided by 37.
- Divisibility by 3 in Three Consecutive Numbers [10/07/2002]
With any combination of consecutive natural numbers, why is one
integer divisible by three and why is ONLY one number divisible by 3?
- Divisibility by 8 [04/14/1997]
Show that, if n is a positive integer, then 5^n+2*3^(n-1) + 1 is
divisible by 8.
- Divisibility by Three: Proof [07/22/2003]
Why is the sum of the digits of a multiple of 3 divisible by 3?
- 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.