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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Lowest Integer That Can't Be Made [04/05/2004]

You have an unlimited number of 'a' cent stamps and 'b' cent stamps, where both a and b are relatively prime positive integers. I am interested in finding a postage value above which any postage amount can be made. I need to find a formula involving a and b, that will give the cutoff point for any such pair of numbers a and b.

Lucky and "Elucky" Numbers in Consecutive Numbers [05/21/1998]

Showing that 13 consecutive numbers always contain at least one number the sum of the digits of which is divisible by 7.

Lucky Number Sequences [05/11/1998]

A lucky number is one for which the sum of its digits is divisible by 7. Can you help me find the following patterns?

Magic Square Matrix [03/24/2003]

Let M be an integer-valued 3x3 matrix whose entries form a magic square. Let s be the sum of all entries in M and d be the determinant of M. Show that d/s is an integer.

Making a Series Sum to Zero [05/24/2002]

How can I place + and - signs between 1^2, 2^2, 3^2, ..., 2005^2 to make the sum equal zero?

Mathematical Induction [01/28/2002]

Use Mathemetical Induction to prove that any postage of at least 8 cents can be obtained using 3- and 5-cent stamps.

Math Virus Formula [10/23/2001]

The virus spreads to all the squares directly touching each other (not including diagonally) and I have found the formula for the number of newly infected cells (although this does not include the first minute)...

Mean of a Set of Numbers by Subsets [08/15/1999]

How can I prove that the mean of a finite set of numbers is the mean of the means of all the non-empty subsets of that set?

Mensa: Numbering for an Alternate World [5/25/1995]

In a parallel universe, the numbering system in use is based on the 26- character Roman alphabet. In this universe you are driving from New York to San Francisco. A road sign indicates you are BBQ miles from San Francisco. Are you closer to...

Minimizing the Sums of Squares [06/12/1998]

Find two numbers such that their sum is 20, and the sum of their squares is as small as possible.

Missile Launch Code [08/03/2003]

What kind of information could you give all 10 people such that if any 3 of them were to get together, they would be able to launch the missiles, but if there were only 2 of them, the information would be insufficient to figure out the code?

Mod [05/17/1999]

What does the term "mod" mean?

Mod Function and Negative Numbers [04/28/2000]

What is the correct value for the function MOD(-340,60)? Microsoft Excel returns the value 20, while Lotus 1-2-3 returns the value -40. Can you explain the difference?

Modular Arithmetic [11/08/2001]

For any integer a, a^4 is congruent to 0 or 1 (mod 5)... We were able to work with the odd number case so it is just the even case that is getting us stuck.

Modular Arithmetic [01/26/2006]

Can you explain why if a = b(mod m), then a = b mod m(mod m), b = a mod m(mod m), and a mod m = b mod m?

Modular Arithmetic and Finding a 13th Root [11/29/2004]

A discussion of a method for easily finding 13th roots based on modular arithmetic.

Modulus algebra: c = ( m * x ) mod p [03/14/2003]

Can anybody tell me an efficient algorithm or solving method to solve the following problem written in Java?

Modulus Congruence Proof [04/18/2001]

How can I prove 2^(3n+2)+21n = 4 mod (49)?

Modulus Operator Problem [2/12/1996]

Can you explain this problem to me? 4 mod 3 = ?

Modulus Proof [04/16/2001]

Can you please show me why m^(2^n) = 1 mod(2^(n+2)) when m is an odd integer?

More on Order of Operations [02/13/2000]

I have found contradictory information on the precedence of the multiplication and division operations. Is there a universally accepted rule for the order of these operations?

Multiple Personality Numbers [03/13/1997]

A rectangular array of a number N is the number of rectangular arrays that can be formed from N dots. Of all the numbers less than 1 million, which has the most rectangular arrays and why?

Multiplication of Integers Modulo (2^16 + 1) [10/18/2002]

Prove that 2^16 * 2^15 mod (2^16 + 1) = 2^15 + 1.

Multiplication of Two Negative Numbers [06/28/2001]

To what extent do the negative numbers we use deserve to be called numbers?

Multiplication Using +, -, and 1/x [10/25/2000]

Suppose you had a calculator with only the +, -, and 1/x operator buttons. Could you do multiplication with it?

Multiplicative Order [08/13/1999]

What is the multiplicative order of 2 mod 2n+1? Can you explain the concept of multiplicative order?

Multiplying by Zero [03/01/2002]

Why does any number times zero equal zero?

((n+1)/2)n [01/31/2002]

If you want to figure out the total of a series of numbers in order, e.g.: 1+2+3+4+5+6+7+8+9 etc., you would use the formula ((n+1)/2)n, where n is the final number of your series. Why?

Narcissistic Numbers, Weird Numbers, and Fortunate Primes [03/27/1998]

Definitions and examples of narcissistic numbers, weird numbers, and fortunate primes.

Natural Logarithms [11/01/1997]

What's "natural" about natural logarithms? Why is 'e' a transcendental number?

Negative Bases [05/13/2002]

How would you represent a number (e.g., 16) in base -2?

Negative Numbers in Binary [08/19/1999]

How do I represent -53 as a binary number? Is it 110101? Wouldn't that be +53?

Negative Numbers to Powers [11/23/2001]

If u is an irrational number and x is a negative number, what is x^u? How do I determine even whether that number is positive or negative?

Nested Square Roots [07/17/1998]

Solve for n where n = sqrt(6 + sqrt(6 + sqrt6 + ...

Newton's Method and Continued Fractions [10/06/1999]

Can you clarify some points on Newton's method of finding square roots without a calculator, and on the continued fraction algorithm (CFA)?

n Factorial - Prove Lower Bound is n^(n/2) [09/11/2001]

I am trying to prove the following: n^(n/2) <= n!

No Integer Solution [10/21/2001]

I don't know how to prove that the following equation has no solution in Z (integers): z^2-2y^2=51.

Non-integer Powers and Exponents [01/06/1999]

How do you find x^n, where n can be an integer, a fraction, a decimal, or an irrational number?

Non-negative Integers [11/15/2001]

How many nonnegative integers consisting of 1-3 digits are divisible by 5? How many nonnegative integers consisting of 1-3 different digits are divisible by 5?

Non-Periodic, Non-Terminating Decimals [06/26/2001]

Why is a non-periodic, non-terminating decimal an irrational number?

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