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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'?

22/7 as an Approximation for Pi [04/01/1998]
Approximating pi by simple continued fractions.

24^25 or 25^24: Proving Which Is Larger [03/12/2015]
An exponent and its base differ by one. Given the two arrangements of the smaller raised to the larger and vice versa, an adult wonders how to compare their relative sizes without actually evaluating them. Doctor Ali applies some offsetting exponentiation en route to a proof via calculus.

36 Sums, Blank Dice [03/15/2001]
You have two blank, six-sided dice, and you can put any numbers on them. The 12 numbers you choose should allow you to make the sums from 1-36...

3 to the Power of 1343 [07/22/2003]
Find the last two digits of 3 to the power of 1343.

400-Digit Product of Two Primes [11/04/2002]
What two prime numbers when multiplied together will equal a 400-digit number?

Absolute Values and Imaginary Numbers [05/17/2000]
Could the solution to |x|= -8 be an imaginary number? Since no absolute value can be negative, this [like sqrt(-1)] cannot be solved.

Abundant Numbers [03/27/1999]
I need to find all the perfect, abundant, and deficient numbers from 1 to 50.

"Accidental Order" in Pi, e [04/12/1999]
Using a result of Dirichlet's to prove that a given sequence will appear in infinitely many prime numbers.

Accountants Use 9 to Check for Errors [02/21/2002]
I know accountants divide the difference of debits and credits by 9 to check for a transposition error. I need to understand why this works.

Adding and Multiplying to Get 7.11 [01/13/1999]
What four monetary values, when added or multiplied, equal $7.11? (Find A, B, C, D, such that A+B+C+D = A*B*C*D = 7.11.)

Adding Hexadecimal Numbers [01/31/2005]
I'm trying to add two hexadecimal numbers, ACF1 + AFFF.

Adding Infinity [06/25/2003]
What is -infinity plus infinity?

Adding Rational and Irrational Numbers [11/07/1999]
How can you prove that a rational number added to an irrational number results in an irrational number?

Algebraic Proof about Product of Twin Primes [04/23/2008]
Show that one more than the product of any two twin primes is always a perfect square.

Algebra Tiles and Negatives [11/05/2001]
Use a model (algebra tiles) to demonstrate that a negative times a negative = a positive.

All Possible Solutions: Diophantine Equations [12/06/2002]
Lance and Mario were working in the snack bar at the Turbulent Tunas Concert. They sold hot dogs for $1.65, hamburgers for $2.35, sodas for $.85, and the combo plate with fries, salad, a hamburger; and soda for $3.89. They sold 80 items in an hour for $163.24. How many of each kind of food did they sell, and how much did they make on each kind of food?

Amicable Numbers [11/12/1998]
What are amicable numbers? Can you give me examples? What's their history?

Amicable Partners [07/23/1998]
Find the amicable partner of 1184. Choose another number between 100 and 1000 and show that it does not have an amicable partner.

Analytical Solution [12/01/1997]
Can you give me an analytical solution of S[N] = Sum[k^2, {k,1,N}] ?

Analyzing Prime Factors [06/14/2007]
Is there a number that has only three prime divisors (3, 5, and 7) and that has a total of 18 divisors?

Analyzing the Results of Large Exponents [09/10/2007]
What is the units digit of 1998^1998? How do I solve problems like this? The number is way too big to calculate.

Angstrom Numbers [02/20/1998]
Numbers (0 and 1 excluded) in which the sum of the cube of the digits is equal to the number itself. Is it true that between the numbers 2 and 10000, there are only 4 such numbers?

Antifirst Numbers [10/23/2000]
An antifirst number is a number with more divisors than every number before it. I need to write a program that will calculate all the antifirst numbers between 1 and 2,000,000,000.

Applications of Different Bases [03/27/2002]
I am trying to find three bases other than base 2, and find a use for them.

Applications of the Number 'e' [02/21/2009]
I've been trying to find examples of real-life applications of the number e, but everything I've found so far just talks about the number itself. How is the number used in the real world?

Applying Euler's Methods [07/27/1999]
Questions about prime divisors, triangle constructions, decomposing quartic polynomials, and rational roots.

Approximating Pi with Continued Fractions [03/18/2006]
Pi is approximated by 22/7. How can you generate increasingly accurate approximations of pi using the division of one integer by another?

Are 0 and 1 Really Rational Numbers? [11/14/2001]
Here's when the laws of rational numbers fall apart: A) 0/1 = 0; B) 0/ 0 = 0 and 1.

Are All Infinitely Long Repeating Numbers Even? [06/06/2000]
Given an infinitely long repeating series, x = 12341234..., then 10000x = 123412341234... Since 9999 is odd and 12340000... is even, can we say that x is even, and therefore all infinitely long repeating series are even?

Are All Perfect Numbers Even? [01/16/1997]
Has it been proved that perfect numbers must be even?

Are Prime Numbers Infinite? [06/18/1998]
How do we know the number of primes is infinite?

Are There Infinitely Many Perfect Numbers? [12/22/1997]
There are so many theories, but nobody seems to have come up with anything definite...

Arithmetic/Geometric Mean Inequality Theorem [04/15/2001]
Prove the AM-GM (arithmetic mean - geometric mean) inequality theorem (prove that (x1+x2+x3+...+xn)/n is greater than or equal to (x1*x2*x3* ...*xn)^(1/n).

Average Age at a Party [10/27/1999]
How can I find b+g if the average age of b boys is g, and the average age of g girls is b, and the average age of everyone, including the 42-year- old teacher, is b+g?

Babylonian Number System [06/15/1998]
A sexagesimal (base 60 instead of 10) number system.

Babylonian Reciprocals (Base 60) [04/08/1999]
What is the reciprocal of the fraction 451/15 ?

Balanced Ternary Notation [04/06/2002]
The place values in a base 3 number system are powers of 3. Suppose the digits are 1, 0 and -1. The base 10 number 35 is written as 110-1 in this base 3 system. Write this base 3 notation for the base 10 numbers 1 through 35...

Bar Codes and Check Digits [10/13/2001]
What do bar codes have to do with math?

Base 16 [07/07/1997]
How do you add and subtract in base 16?

Base 2, Base 8 Multiplication and Addition [10/28/2001]
When to carry: multiplying in bases other than 10.

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