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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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Significance of Irrational Numbers [08/23/1999]

What exactly is the meaning of .333... or pi? What's the difference between point three repeating and point three to the 105th decimal place?

Simple Example of Ramanujan's Work [03/28/1999]

Ramanujan's contributions to the divisibility properties of partitions of whole numbers.

Simultaneous Modulus Congruencies [04/18/2001]

How can I find x if x = 3 (mod 8), x = 11 (mod 20) and x = 1 (mod 15)?

Sizes of Infinities [01/31/1997]

How can you prove that one infinity is larger than another?

Smallest Number Puzzle [06/30/1998]

Find the smallest number which when divided by 9,13,17, and 25 leaves remainders 1,0,2, and 3 respectively.

The Smallest Number Which When Divided Leaves Specific Remainders [02/27/2010]

Doctor Rick, an eleven year-old, and her father apply least common multiples, modular arithmetic, and the Chinese Remainder Theorem to reason their way to the smallest number which when divided by 3, 7, and 11 leaves remainders 1, 6, and 5, respectively.

A Solution in Natural Numbers [10/30/2001]

Prove that x^2+y^2=z^n has a solution in natural numbers for all n, where n is a natural number.

Solving a Diophantine Equation [05/01/2005]

How can I find all integer solutions of an equation in the form aXY + bX + cY + d = 0? For example, 5XY + 3X - 8Y - 8 = 0.

Solving a Diophantine Equation [01/28/2005]

Solve the following Diophantine equation: 5x + 3kx = 8k^2 - 25

Solving Cubic and Quartic Polynomials [04/30/1998]

Could you describe the algorithms used to solve cubic and quartic polynomials (Tartaglia's Solution)?

Solving Diophantine Equations By Organized Thinking [11/25/2003]

Laura is in charge of lighting. Each light fixture supplies exactly 1,000 watts of power to light the bulbs in the fixture. Laura can use any combination of 150-watt, 100-watt, 75-watt, or 60-watt bulbs, but the total number of watts must be 1,000. How many different combinations of bulbs could Laura use in a light fixture?

Solving Modular Formula [11/04/2008]

If a = b^e (mod c), how do I solve for b if I know a and c?

Solving Multivariable Diophantine Equations [05/03/1998]

Finding general solutions to two diophantine equations.

Solving the Diophantine Equation x^y - y^x = x + y [04/30/2005]

Find all integer solutions of x^y - y^x = x + y.

Solving x^y = y^(x - y) for All Natural x, y [11/18/2010]

A student seeks all natural numbers x, y such that x^y = y^(x - y). With chains of reasoning about integer divisibility and exponentiation, Doctor Vogler deduces all three solutions.

Spacing between Prime Numbers [11/08/2005]

Where is the first place that the difference between two consecutive prime numbers exceeds 2000? Is there a formula or general approach to finding such differences without having to just read through lists of known primes?

Splitting a Sum of Integers into Two Equal Sums [09/07/2004]

Given integers {1,2,3,4,...n}, prove that if their sum is even they can be split into two equivalent sums, each equal to half of the original sum, and using each integer once.

The Square and Multiply Method [08/31/1998]

Solve an encryption problem by solving the math function 33815^(81599) (mod 154381).

Square Numbers, All Digits the Same [06/17/2003]

Is there any square number with all the same digits?

Square of an Odd Number [11/12/2002]

True or false: the square of any odd number can be represented in the form 8n+1, where n is a whole number.

Square Root and Sum of Digits [07/25/2008]

I noticed that 81 has the same square root as the sum of its digits since 8 + 1 = 9 and the square root of 81 is also 9. Are there other numbers that have the same property?

Square Root of 2 as a 'Vulgar Fraction' [05/04/2001]

Can the square root of 2 be expressed as a fraction?

Square Root of a Prime [07/14/1999]

Suppose p is a prime number. Show that sqrt(p) is irrational.

The Square Root of i [05/25/1997]

What is the square root of i?

The Square Root of n! [10/14/1998]

For what natural numbers n is the square root of n! an integer?

Square Roots in Binary [10/03/2000]

Can you show an example of taking the square root of a binary number?

Squares in an Infinite Factorial Series [11/23/2001]

How many perfect squares appear among the following numbers: 1!, 1!+ 2!,1!+2!+3!,...1!+2!+3!+...n!?

Square Triangular Numbers [11/22/2002]

Is there an equation for square triangular numbers?

Stirling Numbers [05/26/1999]

Can you show how to evaluate Stirling Numbers of the first and second kinds?

Stirling Numbers of the Second Kind, Bernoulli Numbers [05/29/2001]

Sk = 1^k+2^K+3^k+...+n^k. Find Sk as a formula.

Stirling's Approximation [05/16/2001]

Is there a way to get the answer to a factorial without having to multiply out all the numbers?

Stones, Prime Powers, Induction Proof [01/23/2001]

A heap of 201 stones is divided in several steps into heaps of three stones each...

Subsets and Greatest Common Divisor [03/26/1999]

A question on subsets and another on greatest common divisor (GCD).

Subsets of Real Numbers and Infinity [08/22/2001]

Am I correct in saying that both the whole number set and the integer set have an infinite number of numbers within them, and therefore are of the same size?

Subtracting Two Numbers of Like Base [10/21/2004]

I'm having trouble with the idea of how to borrow when I am subtracting two numbers in a base other than base ten. Can you help?

Subtraction Puzzle [08/18/2002]

For numbers A, B, C, and D, subtract A from B, (or vice-versa; you must be left with a whole number, not a negative one). Repeat with B and C, C and D, and D and A. After about 6 steps, you will always end up with 0000. The puzzle is to get as many steps as possible.

Subtraction Using Nine's and Ten's Complements [05/27/2000]

How does subtraction using the "method of complements" work? Why does it give the correct answer all of the time?

Summing a Binary Function Sequence [07/16/1998]

How do you compute the sum of B(n)/(n(n+1)) from 1 to infinity, where B(n) denotes the sum of the binary digits of n?

Summing Activity Leads to a Mean of e [04/01/2005]

I asked my students to keep adding random integers from 1 to 100 until the sum exceeded 100. We then found the average number of terms added. The answer seems to be e. Why? The more we do it, the closer we get.

Summing a Series Like n*(n!) [10/28/2001]

How can I add up a series like 1*1! + 2*2! + 3*3! ... n*n! ?

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