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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Non-terminating Decimal Representations of Fractions [11/10/1997]

Why when you take a finite, limited quantity like one-third and turn it into a decimal do you get .333... on into infinity?

Normal Numbers [10/19/1999]

Is there a non-probabilistic proof of the existence of normal numbers? Is there an algorithm to produce a number r that is normal in two or more different bases? What about irrational numbers like pi, e, or the square roots of 2,3,5, ...?

No-Solution Equations [11/09/2002]

For any positive integer n, let S(n) denote the sum of its digits. Show that the equation n + S(n) = 1,000,000 has no solution. Then solve the equation n + S(n) = 1,000,000,000.

No Solution: y^2 = x^3 + 7 [03/17/2003]

Show that y^2 = x^3 + 7 has no integer solution.

Number and Its Square Using All 9 Digits Exactly Once [05/22/1998]

Using multiplication facts to find all the whole numbers for which the number and its square together use exactly nine digits 1, 2, 3, ..., 9 only once.

Number Bases beyond 36 [06/22/2003]

What characters are used to represent numbers in a base greater than 37, i.e. beyond Z?

A Number Digits Puzzle [02/23/2001]

How can I determine all positive integers with the property that they are one more than the sum of the squares of their digits?

Number of Digits in a Factorial [09/18/2005]

Is there a way to determine the number of digits that result from a factorial? For instance, 3! = 6 so there is 1 digit and 4! = 24 so there are 2 digits. But what about something much larger, like 125!?

The Number of Divisors of an Integer [04/02/1998]

Formula and proof for the total number of divisors of any integer.

The Number of Zeros in a Factorial [10/01/1998]

How many zeros come after the last non-zero digit of 20,000,000! ?

Numbers and Digit Sums [12/03/1997]

How many numbers between 0 and 99,999 are there whose digits add up to 20?

Numbers in the Fibonacci Sequence [07/19/2001]

How can I show that there is a number in the Fibonacci sequence that ends in 999999999999 ? For what numbers n is there a number in the Fibonacci sequence that ends in n of 9 ?

Numbers Raised to the Negative Power [11/14/2001]

I know that 5^(-N) = 1/5^N. I would like to know why.

Numbers with 12 Factors.... [11/15/1996]

I have to find two numbers that have exactly 12 factors...

Numbers with the Digit 3 [10/27/1999]

In how many numbers between 1000 and 9999 does the digit 3 occur?

Number Theory - Perfect Square [5/26/1996]

Find all the possible values of n...

Number Theory: Primes [07/10/2003]

Find all primes p,q such that ((7^p-2^p)(7^q-2^q))/(pq) is an integer.

Number Theory Proofs [06/24/1999]

How can I prove that the equations (x,y) = g and xy = b can be solved simultaneously if and only if g^2|b for integers g, b?

Numerically Equal Volumes and Surface Areas [06/04/2001]

Find all rectangular solids with integral dimensions, the volumes and surface areas of which are numerically equal.

Odd Digits of Square Numbers [02/07/1999]

Why are there no square numbers other than 1 and 9 that consist entirely of odd digits?

Odd Numbers and Modulo 8 [09/06/2004]

Is it true that any odd number greater than 1, multiplied by itself, is congruent to 1 modulo 8? If so, why?

Odd Perfect Numbers [01/23/2001]

Is this a proof that there are no perfect numbers?

Odd Primes and Primitive Roots [11/14/2001]

Given distinct odd primes P and Q, prove that N = PQ has no primitive roots.

The Official Euclidean Algorithm [11/16/2000]

Can you state briefly the "official" Euclidean Algorithm?

One equals Two [07/25/2001]

There is an algebraic manipulation involving division by zero that results in one equals two, or some other contradiction. What is it?

One Plus One isn't Two [1/10/1995]

I was once shown that 1+1 isn't 2, and I don't remember how it was done. Could you please e-mail me with an answer?

One to the Power of Infinity [07/03/2001]

An instructor says that one to the infinite power does not equal one. If this is true, is there a relatively simple explanation?

Operations in Nondecimal Bases [10/16/1999]

How can you subtract, multiply, and divide numbers in other bases?

Ordering Real Numbers [12/11/2000]

Can you describe how to order real numbers?

The Origin of Lucas Numbers [10/08/1998]

I need help with Lucas Numbers - how and why they were created.

Paint Formulas in Base 48 [08/20/1998]

I work for a paint store where our formulas are based on an ounce being 48 parts...

Pairs of Integers [08/16/1997]

Show that there are infinitely many pairs of integers(x,y) such that x|y**2+m and y|x**2+m where m is any chosen integer; moreover gcd(x,y)=1.

Paradox in the Zero Power [09/26/2000]

Why does the product of multiplying a number by itself zero times equal 1?

Partitioning an Integer [11/14/1998]

How many different ways are there of making a number by adding different combinations of three numbers?

Partitions and Products [01/02/2003]

What is the best way of dividing an integer into parts so that the product of the parts will be as large as possible? Is there a universal law that tells us what partition will produce the maximum product for any given number? And can such a law be proved?

Pascal's Triangle and Fibonacci Formula [02/23/2002]

Prove that the diagonals of Pascal's triangle are the Fibonacci numbers.

Paths to Triangle Points [05/26/1999]

How can I find the number of paths to a point using Pascal's triangle?

Pattern in Period [01/31/1999]

Is there any pattern in odd periods with 1 as numerator?

Patterns in Repeating Decimals [08/06/2003]

Why do certain number sequences repeat in the decimal expansions of common fractions?

Patterns in Rolling 3 Dice [09/16/1997]

I have come out with 216 outcomes when rolling 3 dice; while I was listing all the outcomes, I began to see a pattern...

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