Ask Dr. Math

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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Base e, In, Log [03/08/2002]

Problems in base e. For example: 3e^(2x-1) = 7, or e^(x+1) = 8.

Base of an Exponential Function [09/15/2001]

Why can't the base of an exponential function be negative?

Base of Roman Numerals [04/07/2001]

What is the base of the Roman numeral system, base 10 or base 5?

Bell Numbers [10/15/1998]

How are Bell numbers generated? What are the first 12 Bell numbers?

Beyond Pythagorean Triples [11/20/2001]

a^3+ b^3+c^3 = d^3 and a^4+b^4+c^4 = d^4: equations that are more difficult than the situation with Pythagorean triples.

Binary Addition (2s Complement) [09/25/2001]

Given a number in binary (10110111), I have to convert it to decimal using 2s complement, then to BCD (binary coded decimal). How does 2s complement work?

Binary Divisibility by 10 [04/07/1999]

How can you tell if a binary number is divisible by 10?

Binary Division and Negative Binary Numbers [02/24/2002]

Suppose you enter 11111000 into a binary calculator. How would it know whether that is to be 248 or -8? How does the computer begin the division process? What does it do when it cannot right shift the bits?

Binary Numbers [12/15/1995]

A student asks for information about using base two.

Binary Operations [04/07/1997]

How do you do binary addition, subtraction, multiplication, and division?

Binary Subtraction [03/25/2002]

I keep getting lost when doing the following: 1011000-110010.

Binary to Hexadecimal [12/08/1998]

Is there a simple way to convert from binary (base 2) numbers to hexadecimal (base 16) numbers?

Binet's Formula and Induction [11/28/2001]

What is induction? Can you prove Binet's formula by induction?

Birthday Calendar Puzzle [08/29/2001]

My question involves a game that I have played with my students for a long time, yet I am always unable to explain to them why the pattern works...

Calculating Large Exponentials with Modular Arithmetic [01/02/2005]

What's the easiest way to determine the units digit of (33)^33 + (43)^43 without having to apply brute force? Is there a shortcut?

Calculating Lengthy Repeating Decimals [10/26/2006]

The fraction 355/113 is used as an approximation for pi. A question about converting this fraction to decimal form leads to an interesting discussion of how to calculate long repeating decimals without having to divide them all the way out one number at a time.

Calculators and Irrational Numbers [05/02/2001]

When I square the square root of 11 on any calculator, I get the answer 11 (exactly). That seems to indicate that the square root of 11 is a rational number, but it's not. Can you explain this?

Can a Number to the Zero Power be -1? [01/25/2002]

My teacher told me that (-3^0) is equal to -1. Is this true?

Casting Out Nines [02/19/1998]

I am trying to find a reference that defines this mathematical operation...

Casting Out Nines and Elevens [09/19/1997]

Why is nine used in proving this math answer?

Casting Out Nines for 2nd Graders [07/05/2004]

Can you explain Casting Out Nines in a way that second graders will be able to understand?

Casting Out Nines - Proof [08/30/2002]

Show that if the order of the digits of a natural number is permutated to form a new number, the difference between the old number and the new number is divisible by 9.

Casting Out Nines to Check Arithmetic [08/26/1999]

Can you explain how to use the 'casting out 9's' technique in a way that a 6th grader can understand?

Catalan Numbers and Probability [05/29/2003]

Twenty persons want to buy a $10 ticket each. Ten of them have a $10 note and others have a $20 note. The person at the ticket counter has no money to start with. What is the probability that the person at the ticket counter will not have a change problem?

Catalan's Conjecture [06/02/2007]

Find all integer solutions a and b such that a^b = b^a + 1.

Chinese Remainder Theorem [06/27/1998]

The teacher has some apples to distribute to her students...

Chinese Remainder Theorem and Modular Arithmetic [07/21/2001]

Professor Carroll tries to divide his class into three groups, but two students are left...

A Circular Massacre [09/25/1998]

Ten thousand sailors are arranged in a circle; starting with the first one, every other sailor is pushed overboard ....

Closed Operations for Negative Irrationals [04/28/2001]

What set of operations is closed under negative irrational numbers?

Coding Pairs of Numbers [10/18/2001]

Using the equation: 1/2 ((a + b)^2 + 3a + b) I have plugged in numbers for a and b and worked it out, but I do not see how that "codes the pair" (a,b) into a single number.

Coefficients in a Trinomial Expansion [04/24/2001]

In the expansion of (a+b+c)^6, what is the coefficient of a^2b^2c^2?

Coins in a Square Array [7/8/1996]

I left some coins on the table in a square array and now there are only two left... prove that the butler lied.

Combinatorics: Ramsey Theory [01/12/1998]

Could you help me with a detailed explanation of the theory and a concrete example?

Comparing Numbers in Different Bases [07/28/1999]

If I am given a number in base 4 and asked if it is bigger or smaller than another number in another base, should I always convert both to base 10?

Complement of a Number [7/10/1996]

What is the method for finding the complement of a number?

Complex Powers [09/28/2001]

How do I show that abs(z^i) is less than exp^pi where z is a complex number not equal to 0?

Compressing Numbers and the Shannon Limit [05/12/2004]

Is it possible to find a formula that takes X (a large number) and creates another expression which equals X but is shorter in length than X? For example, take 390,625 which uses six characters. I need a formula which calculates something like 5^8, which equals 390,625 but is only three characters instead of six.

Computing a^((N-1)/2) mod N [12/07/1998]

Is there a shortcut for doing a^((n-1)/2) mod N?

Congruence of Integers [08/10/1998]

Can you help me find the remainder when 5 to the power of 1001 is divided by 6...?

Conjecture About Squares of Consecutive Numbers [08/28/2001]

When any two rational numbers whose absolute values are 1 apart are squared, the difference of the squares is equal to the absolute value of the sum...

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