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
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Selected answers to common questions:
Infinite number of primes?
Testing for primality.
What is 'mod'?
- Group Sizes and Remainders [05/19/2002]
A farmer can divide his sheep into equal groups of 17; but for any
group size less than 17, he gets a remainder of one less than the
group size. How many sheep does he have?
- Guessing a Mystery Number [05/23/2007]
One person thinks of a natural number and the other person has to try
to guess it by asking questions. The one who knows the number will lie
once at some point, but otherwise be truthful in answering the
questions. What's the best algorithm for finding the number, and
what's the fewest guesses it would take?
- Happy Numbers [06/21/1998]
What are happy numbers?
- Happy Numbers [01/05/2006]
What is a "happy" number? What are all the two-digit "happy" numbers
between one and one hundred that are multiples of four?
- Help with Proofs [10/16/1997]
Could you please help me with some proofs from a course called
- Hexadecimal Subtraction and Multiplication [04/06/2000]
How do you subtract or multiply hexadecimal (base 16) numbers?
- Hexadecimal System [02/15/1998]
I know the binary system using base two, but I don't understand the
hexadecimal system using base 16.
- History of Properties [12/11/2001]
Who invented the properties? For example, the distributive property and
the associative property.
- House of Cards [12/02/2001]
Is there a rule for working out the number of cards you need to build a
house of cards of any size?
- How are Binary Codes Used? [01/25/2001]
I've figured out how the binary system works and how to 'translate' from
binary to decimal codes, but how are binary codes used?
- How Can a Number Raised to the Zero Power Be One? [01/23/2008]
I know that 3^7 * 3^0 is 3^(7+0) or 3^7. That means that 3^0 must have
a value of 1. But how is that possible?
- How Does Base 4 Work? [11/06/2003]
How does base 4 work?
- How Many Mice, Cats, and Dogs? [01/13/2003]
You must spend $100 to buy 100 pets, choosing at least one of each
pet. The pets and their prices are: mice @ $0.25 each, cats @ $1.00
each, and dogs @ $15.00 each. How many mice, cats, and dogs must you
- How Many Primes Are Known? [01/21/2003]
How many prime numbers are currently known?
- How Many Rectangular Solids in a Cube? [09/13/2001]
Is there any standard way of finding out how many different possible
rectangular solids can fit into an 3^3 cube?
- How Many Triangles? [10/30/2001]
If we join one point on each of the three sides of a triangle to make
another triangle, there are three triangles with vertices pointing up.
How many triangles will have vertices pointing up if there are n points?
- A Hundred-Row Number Pyramid [11/19/1998]
Starting with two(1,2) in the first row of a pyramid and adding one more
as you go down the list, what is the last number on the righthand side in
the 100th row?
- A Hundred Thousand Switches, One Defect: How Few Tests? [04/19/2011]
Seeking to minimize the binary tests necessary to identify a manufacturing defect, a
student imagines representing the problem as a two-dimensional array, and
pursuing that strategy into higher dimensions. Doctor Anthony introduces a bit-wise
approach for determining the least number, then proceeds through a smaller example
to demonstrate the general method for solving such problems of efficiently identifying
the one bad apple.
- Identity Element [10/12/2001]
What is an "identity element"?
- If n^2 is Even, n is Even [02/21/2002]
I have to show that if n^2 is even then n is also even.
- If N is Odd [12/05/1997]
Prove that if n is odd, then 8 divides (n^2-1).
- Imaginary (Complex) Numbers [02/26/1998]
What are imaginary numbers?
- Incommensurable Numbers [07/25/2001]
What is an incommensurable number?
- Inconstructible Regular Polygon [02/22/2002]
I've been trying to find a proof that a regular polygon with n sides is
inconstructible if n is not a Fermat prime number.
- Increasing and Decreasing Subsequences; Pigeonhole Principle [03/21/2000]
How can I prove that there exists an increasing OR decreasing subsequence
of length n+1 or more in any list of (n^2)+1 distinct integers?
- Indeterminate Forms [09/18/1997]
What is infinity divided by infinity?
- Indeterminate Forms [04/23/2001]
Concerning the indeterminate forms such as 0/0 and infinity/infinity, why
is one to the infinite power considered an indeterminate form?
- The Indeterminate Nature of 0/0 [12/21/2000]
I have a theory that 0/0 = any number, and is not "indeterminate" as is
traditionally claimed. Can you explain the flaw in my thinking, and the
"indeterminate" nature of 0/0?
- Indirect Proofs [01/30/1997]
Give a proof that if r is any nonzero rational number, and s is any
irrational number, then r/s is irrational.
- Induction on .999... [10/19/2000]
In the FAQ proof that .999... = 1, how can you multiply .999... by 10 if
you can never get to the furthest right value? Can you show me an
induction proof that this works?
- Induction Problem [10/14/1997]
Use math induction to prove that (1+2+3+...n)^2 = 1^3+2^3+3^3...n^3.
- Induction Proof of Series Sum [02/03/2001]
How can I prove that for all n greater than 2, the sum 1/(n+1) + 1/(n+2)
+ .. + 1/(2n) is greater than or equal to 7/12?
- Induction Proof with Inequalities [07/03/2001]
Prove by induction that (1 + x)^n >= (1 + nx), where n is a non- negative
- Induction With Binomial Coefficients [10/16/2000]
Prove that the sum from i = 1 to n of (i+k-1 choose k) equals (n+k choose
- Inductive Proof of Divisibility [06/25/2002]
How do you prove that for any integer n the number (n^5)-n is
divisible by 30?
- Inequality Proof for Greatest Integer [10/27/2001]
If x is an arbitrary real number, prove that there is exactly one integer
n that satisfies the inequalities n equal or greater than x less than
- Infinite Continued Fraction [05/15/2002]
What can you determine about the value of the infinite continued
- Infinity as a Skolem Function [10/28/2000]
Is infinity an absolute concept, a relative concept, or both?
- Infinity Hotel Paradox [09/15/1999]
How can a hotel with an infinite number of rooms, all already occupied,
accommodate the passengers of an infinite number of buses without
doubling them up?
- Infinity Solution [04/12/2001]
Can infinity be the solution to the equation 1 + 2x = 3 + 2x?