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'?
- Last Digits and High Exponents [09/26/1998]
Calculate the following expressions without a calculator: 3^1000 and 7^
134. In each, what is the final digit?
- Last Four Digits of 5^64 [03/27/2001]
How can I find the last four digits of 3^125 or 5^64?
- Last Four Digits of the Fibonacci Numbers [05/06/2001]
Show that there is a number ending with four zeros in the Fibonacci
sequence; prove that the Fibonacci sequence has a cycle for the last four
digits with a length of 15,000.
- Least Common Multiple [10/26/1999]
What is the smallest number that is divisible by the numbers 1 through
- The Limit of (1+1/x)^x As x Approaches Infinity [02/17/1998]
How Euler calculated e, and what it has to do with the equation
- Limits of Understanding Division by Zero [09/16/2016]
A parent disagrees with her daughter's teachers about the result of division by zero.
Invoking the floor function and comparing two similar limits, Doctor Ali closes the case
- Line and Unit Circle; Pythagorean Triples [04/16/2001]
If (X,Y) is a point in the 1st quadrant on the unit circle and m is the
slope of the line passing through (X,Y) and the point (0,-1), how can I
express the coordinates (X,Y) in terms of m? Can this be used to generate
- Linear Diophantine Equations [11/27/1997]
... how was the t-variable introduced, and what is the general method?
- Linear Proof [11/07/2001]
We say that f is linear provided that for every x, y in its domain,
f(x+y) = f(x) + f(y). Show that if f is linear and continuous on R (the
set of real numbers), then f is defined by f(x) = cx for some c belong to
- Linear Recurrance Relations [08/10/2004]
Is there a general approach to taking a pattern that is defined
recursively and finding an explicit definition for it?
- Logarithms and Base E [11/20/1997]
Why is the base of a natural logarithm "e" - how did "e" receive a
value of 2.17... ?
- 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
- Modulo a Divisor ... that's Negative? [04/02/2012]
A student evaluates most modular arithmetic expressions without difficulty, but
struggles to make sense of negative divisors. After distinguishing between how
mathematicians and computer scientists interpret mod, Doctor Vogler confirms that
mod(m) and mod(-m) mean the same to mathematicians, then introduces least
- 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