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Browse High School Number Theory
Stars indicate particularly interesting answers or good places to begin browsing.

Selected answers to common questions:
    Diophantine equations.
    Infinite number of primes?
    Testing for primality.
    What is 'mod'?

Uncountable Numbers [10/20/1997]
Why are the real numbers between 0 and 1 uncountable?

Undefined Fractions [02/19/2002]
Why is a fraction with a denominator of zero called "undefined"?

Unique Decomposition of Pythagorean Primes [05/19/2002]
Is it true that a Pythagorean prime (i.e., a natural prime that can be expressed as a sum of squares of two integers) can be expressed as a sum of two squares in one and only one way?

Unknown Numbers and a Venn Diagram [11/26/2001]
The GCF of two numbers is 20 and the LCM is 840. One of the numbers is 120. Explain how to find the other number and use the Venn diagram method to illustrate.

Unsolvable and Unsolved Problems [02/19/1998]
What's the difference between problems like Squaring the Circle and Goldbach's Conjecture or the Collatz Problem?

Uses of Bases Other Than Base 10 [09/28/2004]
Why are there different math bases and what would they be used for?

Uses of Imaginary Numbers [03/24/1997]
Can you tell me a real-life application of imaginary numbers?

Using a Calculator in Other Bases [05/16/2000]
Can a calculator be used to add non-decimal numbers? For example, 27 octal + 65 octal.

Using Algebra to Find Lucky Numbers [12/26/2007]
A lucky number is a positive integer which is 19 times the sum of its digits. How many different lucky numbers are there?

Using Binomial Expansion to Evaluate [2 + sqrt(3)]^50 [11/29/2006]
I've used a computer to evaluate [2 + sqrt(3)]^50 and the answer is extremely close to being an integer. I've tried various expansions of the expression to try and determine why it's so close to an integer, but haven't gotten anywhere. Do you have any idea why?

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 the exponential.

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 are they?

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 indeterminate forms.

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.

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