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High School Archive

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'?

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22/7 as an Approximation for Pi [04/01/1998]

Approximating pi by simple continued fractions.

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.

Base Conversions: Trial and Error or Formula? [11/26/2002]

Find base x of a given number that is equivalent to a number in any base.

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