See also the
Dr. Math FAQ:
0.9999 = 1
0 to 0 power
n to 0 power
0! = 1
dividing by 0
Browse High School Number Theory
Stars indicate particularly interesting answers or
good places to begin browsing.
Selected answers to common questions:
Infinite number of primes?
Testing for primality.
What is 'mod'?
- Long Division in Base 36 [11/05/2003]
I am trying to use an 'imaginary' system that is base 36 (ie. 1, 2, 3,
4, 5, 6, 7, 8, 9, A=10, B=11, ..., Z=35) and I am struggling to divide
one number in this system by another using long division.
- Long Division in Binary [05/16/2000]
How can you divide 1011 base 2 by 11 base 2?
- Long Division in Binary [06/06/2000]
What is the algorithm for binary division? Can you show me some examples?
- 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)...
- Maximizing Irregular Polygon Area: Which Circle? [05/01/2011]
How do you determine the radius of the circle that maximizes the area of an irregular
n-gon circumscribed on it? With the Pari computer algebra system, Doctor Vogler
approaches the question using numerical techniques such as Newton's Method and a
binary search, which suggests that no closed-form expression exists.
- 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
- 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
- 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
- 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
- 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
- Natural Logarithms [11/01/1997]
What's "natural" about natural logarithms? Why is 'e' a transcendental
- 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
- 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!