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'?
- Even or Odd in Base 5? [09/23/1999]
Is there a way to find whether a number written in base 5 is even or odd
without first converting it to base ten?
- Explaining the Euclidean Algorithm [10/27/1998]
In the Euclidean Algorithm (or the Division Algorithm), why is the last
divisor the greatest common factor?
- Exponential Diophantine Equation [06/24/2005]
Find three integers a,b,c > 1 such that a^a * b^b = c^c.
- Exponential Proof [03/06/2003]
Let a, b, c be positive integers such that a divides b^2, b divides
c^2, and c divides a^2. Prove that abc divides (a + b + c)^7.
- Exponential Series Proof [05/05/2001]
Given e^x greater than or equal to 1 + x for all real values of x,and
that (1+1)(1+(1/2))(1+(1/3))...(1+(1/n)) = n+1, prove that e^(1+(1/2)+
(1/3)+...+(1/n)) is greater than n. Also, find a value of n for which
1=(1/2)+(1/3)+...+(1/n) is greater than 100.
- Extending the Extended Euclidean Algorithm? [04/06/2017]
A young adult wonders about applying the Extended Euclidean Algorithm to solve
Diophantine equations in more than two variables. Doctor Jacques shows her how,
working through an example step by step, before introducing an alternative approach.
- Factorial Base and Base 10 [11/02/2001]
Let n be a number written in base 10, which also has an interpretation in
factorial base. Let m be the value of its interpretation in factorial
base. What is the greatest n for which m is equal to or less than n?
- Factorials Can't Be Squares [02/11/2000]
Can you prove that the factorial of a number (greater than 1) can never
be a perfect square?
- Factoring [02/09/1999]
Find the smallest number (integer) that has 30 factors.
- Factoring 13 with Complex Numbers [08/11/1998]
How do you show that 13 is not prime using imaginary numbers? We know
that 13 = (3 + 2i)(3 - 2i), but how do you do this in general?
- Factoring Large Numbers [10/26/1998]
Can you give me an algorithm for factoring large numbers? What about the
Pollard Rho Factoring Algorithm?
- Factoring Large Numbers [05/26/2000]
How can you use Fermat's Little Theorem to factor large numbers?
- Farey Series [10/21/2002]
For three successive terms in a Farey's series, say a/b, c/d, e/f, how
can we prove independently that c/d = (a+e)/(b+f) and ad-bc = -1 ?
- Fermat Number Proof [01/30/2001]
Prove that if n is greater than 0, then the Fermat number 2^2^n + 1 is of
the form 9k-1 or 9k-4. Prove that n and 2^2^n + 1 are relatively prime
for every n greater than 0.
- Fermat's Factorization Method [01/29/1999]
Can you describe Fermat's method of factoring an integer?
- Fermat's Last Theorem for n = 3 [12/14/1998]
What is the proof for Fermat's Last Theorem where n = 3? Who is given
credit for the first proof for this case?
- Fermat's Last Theorem with Negative Exponents [10/26/2000]
Are there any solutions of Fermat's Last Theorem, x^n + y^n = z^n, for n
less than 2?
- Fermat's Little Theorem [09/02/2000]
Can you help me prove Fermat's Little Theorem, that the expression n^p-n,
where p is an arbitrary prime and n is a positive integer, is always
divisible by p?
- Fermat's Little Theorem and Prime Numbers [09/28/1998]
Please explain how to use Fermat's Little Theorem to test whether a
number is composite.
- Fermat's Little Theorem: A Special Case [06/26/2001]
Show that n^7-n is divisible by 7.
- Fermat's Theorem [01/21/1998]
Why was Fermat's Theorem such a mystery?
- Fibonacci Formula Inductive Proof [11/05/1997]
I must prove by induction that F(n) = (PHI^n - (1 - PHI)^n) / sqrt5...
- Fibonacci-GCD Proof [11/20/2002]
Can you help me prove that fib(gcd(m, n)) = gcd(fib(m), fib(n)) ?
- Fibonacci Identity [12/10/2001]
I am trying to create an inductive proof for the particular identity of
Fibonacci numbers that: F(n-1) * F(n+1) = (-1)^n + (Fn)^2.
- Fibonacci or Lucas Number [02/19/2003]
How do I know that any number x is a Fibonacci or Lucas number?
- Fibonacci Proof [01/29/2001]
This proof is giving me major problems: F(2n) = (F(n))^2 + (F(n-1))^2.
- Fibonacci Sequence [01/29/2001]
Is there a formula for the n-th Fibonacci number?
- Fibonacci Sequence Property [11/29/2001]
I have to prove that in the Fibonacci sequence, F(k) is a divisor of
F(nk), where n is a natural number (so, F(nk) = A*F(k) where A is a
- Fibonacci Sequences [01/08/1998]
Please help me with a proof.
- Fibonacci's Liber Quadratorum - Proposition 18 [04/07/2002]
Prove by contradiction that if any two positive integers have an even
sum, then the ratio of their sum to their difference will not be the
same as the ratio of the larger number to the smaller.
- Find a, b, c, Such That a! b! = a! + b! + c! [12/09/2003]
Find all triples of nonnegative integers a, b, c such that a! b! = a!
+ b! + c!
- Finding 13^99 [11/21/2001]
What is the units digit of 13 to the 99th power?
- Finding a Desired Perfect Cube [04/26/2007]
What is the smallest positive cube that ends with the digits 2007?
- Finding A Number Given Its Divisors and Remainders [10/22/2003]
A general strategy for solving problems such as finding the smallest
whole number that when divided by 5, 7, 9, and 11 gives remainders of
1, 2, 3, and 4 respectively.
- Finding a Number Given the Sum of Its Factors [10/19/2007]
The factors of an unknown number add up to 91. Is there a way to find
the number without having to just use trial and error?
- Finding a Remainder [09/21/2007]
When the even integer n is divided by 7, the remainder is 3. What is
the remainder when n is divided by 14?
- Finding a Series Given the Sum [09/27/1999]
How can I find all series of consecutive integers whose sum is a given
- Finding a Set of Consecutive Odd Integers That Sum to a Given Number [02/21/2004]
Given a number n, which is the sum of some set of consecutive odd
numbers, is there an efficient way to find the set of odds that
generate the sum?
- Finding Catalan Numbers [12/15/1999]
What are Catalan numbers and what applications do we have for them?
- Finding Divisibility Rules for Large Numbers [12/21/2000]
Is there any system for finding divisibility rules for any number?