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
good places to begin browsing.
Selected answers to common questions:
Infinite number of primes?
Testing for primality.
What is 'mod'?
- Remainder when Dividing Large Numbers [04/17/2001]
How can I find the remainder when (12371^56 + 34)^28 is divided by 111?
- Repeating Decimals [04/28/1999]
I am interested in finding longer repeating groups in number tails of
- Repeating Decimals - Rational or Irrational? [09/11/2001]
Are 0.252252225... and 0.125126127... rational or irrational?
- Repeating Digits of Fractions [04/28/1999]
Do you know any theorems relating to the length of the repeating portion
of the decimal representation of fractions?
- Repetend Runaround [08/29/2017]
A twelve year-old struggles to manually determine the digit length of a repetend.
Doctors Greenie and Peterson explain the challenge — and share a shortcut.
- Reversal of Age Digits Every Eleven Years [11/06/2007]
Every 11 years, my age is the exact reverse of my mother's age. When I
was 13 she was 31, when I was 24 she was 42, and so on. Why does this
- Reversed Digits Theorem [06/24/1999]
For a positive integer abc..., if (abc...)^n = xyz... and if
(a+b+c+...)^n = x+y+z+..., how can I prove that (...cba)^n = ...zyx?
- Reverse Modulus Operator [10/09/2001]
Is there an operator that would return 2 when we we do 6 * 0, * being
this new operator?
- Roots of ax^f2 bx+c = 0 [05/22/1997]
Prove that if a,b,c are odd integers, then the roots of ax^2 bx+c=0 are
- Rounding Negative Numbers [05/23/2007]
I was wondering what -0.5 is rounded to the nearest integer?
- RSA Encryption [04/25/2002]
Decrypt the encrypted message in ciphertext C to find the original
plaintext, a string of English letters.
- Running Time of an Insertion Sort [08/06/1999]
How can I write and solve a recurrence formula for the running time of an
insertion sort? Which is better, an insertion sort or a merge-sort?
- Search for the Largest Prime [08/01/2000]
What is the largest finite number that has a practical use in some branch
of mathematics or science? What is the largest prime number known?
- Second-Degree Two-Variable Diophantine Equation [04/12/2001]
Solve Ax^2+Bxy+Cy^2+Dx+Ey+F = 0 where B^2-4AC=k^2 for some integer k.
- Second-Order Linear Recurrences [06/08/2001]
Three problems involving recurrence equations.
- Second Order Recurrence with Non-Constant Coefficients [05/27/2005]
I'm trying to find a closed form solution of a second order recurrence
relation with no constant coefficients, specifically: u(n+2) =
2*(2*n+3)^2 * u(n+1) - 4*(n+1)^2*(2*n+1)*(2*n+3)*u(n). Can you help?
- Semitonal Half-Stepping, Ever Sharper [05/05/2012]
An octogenarian wonders why modulating through the circle of fifths results in adjacent
scales that become successively sharper. Doctor George applies some modular
arithmetic to peel away at the chromatic scale — then spontaneously augments
his original response to offer clearer insight.
- Sequence of Integers [08/12/2008]
Find all functions f such that for each n in Z+ we have f(n) > 1 and
f(n + 3)f(n + 2) = f(n + 1) + f(n) + 18.
- Set Theory and GCD and Divisibility [01/27/2003]
If 1 <= a <= n and 1 <= b <= n and ab <= n, and if a divides n and b
divides n, does that mean that ab divides n given that GCD(a,b) = 1 ?
- Show 2^(N-1) Congruent to 1(mod N) [02/25/2003]
I need to show that if N = 2^p - 1, p prime, then 2^(N-1) is congruent
to 1(mod N).
- Showing a Diophantine Equation Has No Solutions [07/30/2008]
Do there exist positive integers m and n such that m^3 = 3n^2 + 3n + 7?
- Showing Divisibility [07/12/1998]
How do you show that 5^(2n) + 3(2^(2n+1)) is divisible by 7?
- Showing Two Numbers Are Relatively Prime [08/01/2008]
Show that for every natural number n, 21n + 4 and 14n + 3 are
- Show n^3 + 11n Divisible by 6 [12/11/2002]
If n is a natural number, show that for all values of n, (n^3+11n) is
divisible by 6.
- Sigma Notation [12/17/1998]
Some summation formulas; finding Sum((n+1)^2).
- Significance of Irrational Numbers [08/23/1999]
What exactly is the meaning of .333... or pi? What's the difference
between point three repeating and point three to the 105th decimal place?
- Simple Example of Ramanujan's Work [03/28/1999]
Ramanujan's contributions to the divisibility properties of partitions of
- Simultaneous Modulus Congruencies [04/18/2001]
How can I find x if x = 3 (mod 8), x = 11 (mod 20) and x = 1 (mod 15)?
- sin(sqrt2) More Simply? [05/17/2017]
A teen seeks simpler forms for trigonometric functions of radicals. With examples,
Doctor Vogler explains how this pursuit leads to infinite cardinalities of sets and further number theory.
- Sizes of Infinities [01/31/1997]
How can you prove that one infinity is larger than another?
- Smallest Number Puzzle [06/30/1998]
Find the smallest number which when divided by 9,13,17, and 25 leaves
remainders 1,0,2, and 3 respectively.
- The Smallest Number Which When Divided Leaves Specific Remainders [02/27/2010]
Doctor Rick, an eleven year-old, and her father apply least common multiples,
modular arithmetic, and the Chinese Remainder Theorem to reason their way to the
smallest number which when divided by 3, 7, and 11 leaves remainders 1, 6, and 5,
- A Solution in Natural Numbers [10/30/2001]
Prove that x^2+y^2=z^n has a solution in natural numbers for all n, where
n is a natural number.
- Solving a Diophantine Equation [05/01/2005]
How can I find all integer solutions of an equation in the form
aXY + bX + cY + d = 0? For example, 5XY + 3X - 8Y - 8 = 0.
- Solving a Diophantine Equation [01/28/2005]
Solve the following Diophantine equation: 5x + 3kx = 8k^2 - 25
- Solving Cubic and Quartic Polynomials [04/30/1998]
Could you describe the algorithms used to solve cubic and quartic
polynomials (Tartaglia's Solution)?
- Solving Diophantine Equations By Organized Thinking [11/25/2003]
Laura is in charge of lighting. Each light fixture supplies exactly
1,000 watts of power to light the bulbs in the fixture. Laura can
use any combination of 150-watt, 100-watt, 75-watt, or 60-watt bulbs,
but the total number of watts must be 1,000. How many different
combinations of bulbs could Laura use in a light fixture?
- Solving Modular Formula [11/04/2008]
If a = b^e (mod c), how do I solve for b if I know a and c?
- Solving Multivariable Diophantine Equations [05/03/1998]
Finding general solutions to two diophantine equations.
- Solving the Diophantine Equation x^y - y^x = x + y [04/30/2005]
Find all integer solutions of x^y - y^x = x + y.