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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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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 buy?

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?

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 integer.

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 k+1).

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 n+1.

Infinite Continued Fraction [05/15/2002]

What can you determine about the value of the infinite continued fraction [1;1,2,3,1,2,3,1,2,3....]?

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?

Infinity to the Zero Power [04/28/2001]

Does (infinity)^0 equal 1? Why or why not?

Integer Iteration Function [12/24/2003]

Let X be a positive integer, A be the number of even digits in that integer, B be the number of odd digits and C be the number of total digits. We create the new integer ABC and then we apply that process repeatedly. We will eventually get the number 123! How can we prove that?

Integer Logic Puzzle [04/22/2001]

Two integers, m and n, each between 2 and 100 inclusive, have been chosen. The product is given to mathematician X and the sum to mathematician Y... find the integers.

Integer Root Checking [02/18/2003]

Is there a quick way to check whether a number has any roots that are whole numbers?

Integers and Complex Numbers [02/27/1997]

Do hyper-reals and octonions exist outside complex numbers?

Integers and Fractions [03/23/2002]

Give an example of a positive integers p,a,b where p/ab and not p/a and not p/b. Let m, n, and c be integers. Show that if c/m then c/mn.

Integer Solutions of ax + by = c [04/03/2001]

Given the equation 5y - 3x = 1, how can I find solution points where x and y are both integers? Also, how can I show that there will always be integer points (x,y) in ax + by = c if a, b and c are all integers?

Integer Solutions to a Cubic Equation [04/11/2005]

Fermat's method of infinite descent is used to show that the cubic equation (a^3) + (2b^3) + (4c^3) - 4abc = 0, with a, b, and c whole numbers and without a=b=c=0, has no solution.

Interesting Diophantine Equation [12/06/2005]

Find all integers x such that x^2 + 3^x is the square of an integer.

Intersection of Lines [06/28/1998]

n co-planar lines are such that the number of intersection points is a maximum. How many intersection points are there? ...

An Introduction to Basic Diophantine Equations [08/27/2007]

A birdcage contains both 2-legged and 1-legged birds, and there are a total of 11 legs in the cage. Use a Diophantine equation to find all possible combinations of birds.

Introduction to Infinity, Limits, and Why 0.999... Equals One [01/12/2008]

I've read the FAQs and other questions sent to you concerning why 0.999... equals one, but I'm still not convinced. I don't entirely understand infinity and think it creates more problems than it solves.

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