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
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Selected answers to common questions:
Infinite number of primes?
Testing for primality.
What is 'mod'?
- Subsets of Real Numbers and Infinity [08/22/2001]
Am I correct in saying that both the whole number set and the integer set
have an infinite number of numbers within them, and therefore are of the
- Subtracting Two Numbers of Like Base [10/21/2004]
I'm having trouble with the idea of how to borrow when I am
subtracting two numbers in a base other than base ten. Can you help?
- Subtraction Puzzle [08/18/2002]
For numbers A, B, C, and D, subtract A from B, (or vice-versa; you
must be left with a whole number, not a negative one). Repeat with
B and C, C and D, and D and A. After about 6 steps, you will always
end up with 0000. The puzzle is to get as many steps as possible.
- Subtraction Using Nine's and Ten's Complements [05/27/2000]
How does subtraction using the "method of complements" work? Why does it
give the correct answer all of the time?
- Summing a Binary Function Sequence [07/16/1998]
How do you compute the sum of B(n)/(n(n+1)) from 1 to infinity, where
B(n) denotes the sum of the binary digits of n?
- Summing Activity Leads to a Mean of e [04/01/2005]
I asked my students to keep adding random integers from 1 to 100 until
the sum exceeded 100. We then found the average number of terms
added. The answer seems to be e. Why? The more we do it, the
closer we get.
- Summing a Series Like n*(n!) [10/28/2001]
How can I add up a series like 1*1! + 2*2! + 3*3! ... n*n! ?
- Summing Consecutive Integers [08/30/1998]
Express 1994 as a sum of consecutive positive integers, and show that
this is the only way to do it.
- Summing n^k [11/24/1998]
Is there a general formula for summing the n^k, where k is a positive
- Sum of 1/Sqrt(i) [11/20/2000]
What is the formula for the sum of 1/sqrt(i) for i = 1 to n? Can you show
me the proof by induction?
- Sum of Consecutive Odd Integers [07/27/2001]
Given an integer N, can N can be written as a sum of consecutive odd
integers? If so, how can I identify *all* the sets of consecutive odd
integers that add up to N?
- Sum of Digits Divisible by 11 [08/16/1999]
Can you prove that in a sequence of 39 consecutive natural numbers there
exists at least one number such that the sum of its digits is divisible
- Sum of Digits of Multiples of Nine [08/12/2004]
Can you prove that if you add the digits of any multiple of nine, then
add the digits of that result, and keep going, you eventually wind up
with 9? For example, 99 => 9 + 9 = 18 => 1 + 8 = 9. Why does it work?
- Sum of Distinct Fibonacci Numbers [05/06/2001]
How do you show that every positive integer is a sum of distinct terms of
the Fibonacci sequence?
- Sum of First n Cubes, First n Squares [11/18/2002]
Is there a shortcut to find (1^3-1^2)+(2^3-2^2)+(3^3-3^2)... (15^3-15^
- Sum of First n Natural Numbers [12/03/2005]
Factorial refers to the product of the first n natural numbers. Is
there a name and symbol for the SUM of the first n natural numbers?
- Sum of Integers [07/03/2001]
How many integers are 13 times the sum of their digits?
- Sum of Numbers from 1 to n [01/09/2003]
Is there a formula for calculating the summation of numbers from 1
- Sum of Powers of 2 [08/28/2001]
I want to derive a formula for the sum of powers of 2.
- Sum of Squares of Two Odd Integers [10/26/1999]
How can I prove that the sum of the squares of two odd integers cannot be
a perfect square?
- Sum of Twin Primes [09/04/2003]
Given that a and b are two consecutive odd prime integers, prove that
their sum has three or more prime divisors (not necessarily distinct).
- Sum of Two Cubes [01/12/2002]
Find the smallest number that can be expressed as the sum of two cube
numbers in two different ways.
- Sum of Two Different Primes [02/22/2002]
Can the sum of two different primes ever be a factor of the product of
- Sum of Two Squares [12/04/1997]
What is the smallest number that can be expressed in twelve different
ways as the sum of two squares?
- Sum of Two Squares [05/26/2003]
Can you generate the sequence [400, 399, 393, 392, 384, 375, 360, 356,
337, 329, 311, 300]?
- Sum of Unit Fractions [07/17/2001]
By induction, prove that every proper fraction p/q with p less than q can
be written as a finite sum of distinct reciprocals of positive integers.
- Sums Divisible by 11 [10/10/2001]
Why is the sum of a number with an even number of digits and that same
number written in reverse always divisible by 11?
- Sums of Consecutive Integers [01/03/2001]
How many different ways can 2000 be expressed as the sum of two or more
consecutive positive integers?
- Sums of Consecutive Integers [02/04/2001]
What numbers can be expressed as the sum of a string of consecutive
- Sums of Consecutive Integers with Digital Sums [01/27/2004]
Find all sets of positive consecutive integers that sum to 100, and
whose digits sum to greater than 30.
- Sums of Consecutive Numbers [06/20/2002]
In what way(s) can 1000 be expressed as the sum of consecutive
- Sums of Consecutive Odd vs. Even Integers [04/15/2002]
Can the sum of two consecutive even integers ever equal the sum of
two consecutive odd integers? Why or why not?
- Sums of Consecutive Positive Integers [03/02/2001]
Why are the powers of 2 the only numbers you cannot get as the sum of
a series of consecutive positive integers?
- Sums of Sets of Prime Numbers [01/07/2003]
Given several sets of prime numbers, use each of the nine non-zero
digits exactly once. What is the smallest possible sum such a set
- Sums of Square Integers Puzzle [07/01/2002]
How many numbers from 1-100 can be expressed as the sum of the squares
of two positive integers?
- Sums of Three Squares [05/18/1998]
What numbers cannot be expressed as the sum of three squares?
- Synthetic Division [11/13/1997]
Why does synthetic division work?
- System-Level Programming and Base 2 [05/03/2001]
In computer programming, I have a result that contains several values,
always a power of 2 (2^2, 2^3, 2^4). If my value is 2^3, 2^4, 2^6 304,
how can I tell if 2^3 exists in 304?
- Systems with More Variables than Equations [12/11/2002]
How do I solve a system that has three variables and only two equations, such as 187y + 98x + 45z = 48 and 2y + 9x + 3z = 198?
- Taylor Expansion [11/21/2001]
Can you give me the proof of this statement: arcsin(x) = x + 1/2 (x^3/ 3)
+ (1/2)(3/4)(x^5/5) + (1/2)(3/4)(5/6)(x^7/7) + ... The basis of the
calculation is a Taylor series.