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'?
- Using Mod to Find Digits in Large Numbers [12/10/1996]
Find the last two digits in 1996^1996.
- Using Modular Arithmetic to Find Remainders [10/21/2004]
What is the remainder when 2^(2^405) is divided by 23?
- Using Modular Arithmetic to Test Divisibility of Large Numbers [08/30/2008]
Prove that 55^62 - 2*13^62 + 41^62 is divisible by 182.
- Using Weighted Criteria to Make Decisions [07/18/2008]
I have three employees who have each worked different numbers of days
and learned different numbers of skills in that time. How can I decide
who is the most effective employee by weighting those two factors?
- USSR Math Olympiad Puzzle [04/16/2003]
Prove that no matter what string you start with, the letters at
the corners of the triangle are either all the same or all different.
To what other numbers could you change the 'string of 10 letters' and
still have the assertion be true?
- Variable Within and Outside an Exponent [07/29/1997]
Solve for t: d = a*t + b*e^-(c*t) where a, b and c are constants and e is
- Was Euler wrong? 2*Pi=0? [03/13/2002]
While I was surfing the Internet, I found a site with an interesting
proof that shows that 2*Pi = 0 by using Euler's famous equation...
- A Way to Think about the Locker Problem [01/16/2004]
Imagine there is an endless string of lockers in your school. Person 1
starts at locker 1 and opens every locker. Person 2 starts at locker 2
and closes every 2nd locker. Person 3 starts at locker 3 and changes
every 3rd locker. Person 4 starts at locker 4 and changes every 4th
locker. Person x starts at locker x and changes every xth locker.
Which lockers are left open in rows of 25, 100, and 500 lockers?
- What Exactly is a Fraction? [10/15/2001]
What is a fraction? Is 3/1 a fraction? Is 5/sqrt(2) a fraction?
- What is 0^0? [11/01/2001]
We are doing exponents in school and we were talking about how 9^0=1,
10^0=1, etc., and I asked what 0^0 is.
- What is 0.999... + 0.999...? [09/27/1999]
How can I show that 0.999... + 0.999... = 2?
- What is Modulus? [06/06/2001]
I have used the mod command and know what the results mean, but I don't
understand the theory behind it and what is actually happening.
- What is Octal? [11/30/1999]
What is octal?
- What is the Gamma Function? What is Gamma of 4? [05/28/1998]
Deriving G(4) = 3! from the gamma function integral.
- What is the Largest Named Number? [12/08/2004]
A discussion of various large numbers and how they are defined,
including google, googleplex, and Moser, Skewes, and Graham numbers.
- What Kind of Number is One? [12/06/2001]
What is one, if it is neither prime nor composite?
- What Type of Number Are Zero and One? [11/08/2004]
If 1 and 0 are not considered prime or composite numbers, then what
- When Casting Out Nines Fails [11/29/2001]
To prove that casting out 9's worked, I intentionally used an incorrect
quotient, but was shocked when the problem checked out correctly.
- Which Fractions Repeat? [07/21/1998]
How do you know whether a fraction will be a repeating or terminating
decimal? If repeating, how many decimal places?
- Why Are 1^infinity, infinity^0, and 0^0 Indeterminate Forms? [05/08/1998]
Using limits to prove that 1^infinity, infinity^0, and 0^0 are
- Why Aren't There Negative Prime Numbers? [12/10/1999]
Why can't negative numbers be prime numbers?
- Why Can't 0 Divided By 0 Be 0? [11/25/2003]
Why can't you divide 0 by 0? I've thought about it and it seems that
dividing zero objects into zero groups will result in zero groups. Why
doesn't this work?
- Why Casting Out Nines Works [10/13/2008]
Can you explain in simple terms why the Casting Out Nines method works?
- Why Does 0^0 = 1 and Not Undefined? [11/30/2007]
Your proof for why x^0 = 1 uses a law which breaks down at x = 0. Then
in your definition for 0^0 you side significantly in favor of 0^0 = 1
based on your rule for x^0 = 1 (which was based on a law that breaks
down at 0). Based on what I've read I would side in favor of
undefined. Are there any more conclusive reasons for siding with 0^0 = 1?
- Why Does 0! = 1 ? [12/8/1995]
Why does 0! = 1 ?
- Why is 0! 1? [09/14/1997]
Why is zero factorial 1?
- Why Use Q and Z? [09/12/2001]
Why is the letter Q used for rational numbers and Z for integers?
- Wilson's Theorem [03/03/2002]
I'm looking for a proof for Wilson's theorem: n divides (n-1)! + 1 if and
only if n is a prime number.
- Would Aliens Use Base 10? [05/13/2002]
If aliens had 6 fingers, would they use base 12?
- Write a Sum That Totals 100 Using Digits 1 - 9 [05/30/2004]
Is it possible to arrange the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9 so that when added they total 100? Only adding is permitted and the numbers can be rearranged. Each number can only be used once.
- Writing Numbers in Bases Greater Than 10 [04/05/2001]
What would 4 x 13 [base 10] look like in base 42? Do all bases above ten
use the same method?
- Zero and Imaginary Numbers [07/18/2001]
Is it true that zero divided by an imaginary number is zero? How could
the answer be in the real number line when the divisor can't be found in
the real number line?
- Zero and Infinity [04/24/1997]
Why is the quotient of a number divided by zero infinity?
- Zero as an Exponent [7/15/1996]
Why does n^0 = 1?
- Zero as Denominator [10/22/1997]
Why can't zero be in the denominator for rational numbers?
- Zeroing in on the Cell, Thinking in Binary [03/04/2013]
Following a simple constraint, a student populates a matrix, resulting in a Sudoku-like
array — but which number is in a given row and column? Doctor Douglas picks
up on a pattern that the student noticed, and extends it by thinking in base two.
- Zero Laws and L'Hopital's Rule [03/04/1998]
Is zero divided by zero: a) zero, b) undefined, or c) one?
- The Zero Power of Two [12/10/1998]
Why is 2 to the 0 power equal to 1? I don't understand how a number can
be multiplied by itself zero times.
- Zeros between 1 and 222 Million [11/17/2001]
How many zeros will I use if I write down all the numbers from 1 to 222
million? And how can I generalize this?
- Zero to a Negative Exponent [05/06/2001]
Is 0^(-3) equal to 0, or is it undefined? We can't determine whether to
use the 0^x = 0 rule, or to interpret it as 1/(0^3).