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Number Theory

These problems require knowledge of the properties of rational numbers, irrational numbers, or real numbers. They may include a need to know the divisibility rules, factorization, prime and composite numbers, sequences or series. They should be suitable for the beginning prealgebra student.
Some of these problems are also in the following subcategories:
 Related Resources

Interactive resources from our Math Tools project:
Math 7: Number Theory
The closest match in our Ask Dr. Math archives:
Middle School Archive
NCTM Standards:
Number and Operations Standard for Grades 68
Access to these problems requires a Membership.

Birthday Line Up
 Claire Mead, Stephen Weimar

PreAlgebra, difficulty level 2. How old will Seth be when his mother is five times Seth's age?
... more>>

Can You Find My Numbers?
 Roya Salehi

PreAlgebra, difficulty level 3. I have eight numbers in my list. Use the clues to find my numbers.
... more>>

Choosing Cs
 Lisa Lavelle, Suzanne Alejandre

PreAlgebra, difficulty level 3. Given the expression c^2  2, what is the probability that the result
is composite for integers 10 through 2?
... more>>

Coding Conundrums
 Mary DeYoung & class

PreAlgebra, difficulty level 2. Explore encoding and decoding in this cipher puzzle.
... more>>

Counting Chicken Wings
 Annie Fetter, Suzanne Alejandre

PreAlgebra, difficulty level 3. Figure out how many wings are in the Biggest Bucket O' Wings.
... more>>

A Cranberry Craving
 Claire Mead, Suzanne Alejandre

PreAlgebra, difficulty level 2. Use the pattern to find out how many cranberries Carissa ate on
Thanksgiving Thursday.
... more>>

Divisible by a Dozen
 Terry Trotter & Annie Fetter

PreAlgebra, difficulty level 3. If you take any three consecutive integers, multiply the first times the square of the second, then multiply by the third, is 12 a factor of the product?
... more>>

Fall Festival Frolic
 Judy Ann Brown

PreAlgebra, difficulty level 3. How many different types of candy are needed to make at least 50 different treat bags?
... more>>

Falling into Sequences
 Judy Ann Brown

PreAlgebra, difficulty level 3. Find the 20th term in a sequence using both a recursive method and an explicit formula.
... more>>

Falling Leaves
 Suzanne Alejandre

PreAlgebra, difficulty level 2. Find the number of leaves that were falling.
... more>>

Fences for Fido and Friends
 Annie Fetter

PreAlgebra, difficulty level 2. Find the area of each rectangular dog pen.
... more>>

How Old is Ms. Nelson?
 Judy Ann Brown

PreAlgebra, difficulty level 2. Ms. Nelson claims that she is three times as old as her little sister. However, six years ago she was five times as old as her sister was. How old is Ms. Nelson?
... more>>

An Interesting Integer
 Judy Ann Brown

PreAlgebra, difficulty level 2. I'm thinking of a particular integer. It is a multiple of three, five, and seven. No digit occurs more than once. Can you find my number?
... more>>

Kate's Combination
 Suzanne Alejandre

PreAlgebra, difficulty level 2. Find Kate's locker combination using mathematical clues. Is there more than one possibility?
... more>>

Lillian's Lines
 Lillian Ray and Molly Ribble

PreAlgebra, difficulty level 3. Students are asked to find how many square panes a path goes through
for a window of a
given size.
... more>>

Lost in Des Moines
 Judy Ann Brown

PreAlgebra, difficulty level 2. Calculate the speed of a car using given information.
... more>>

Lucky Lollipops
 Suzanne Alejandre & Maria Fung and her students

PreAlgebra, difficulty level 3. Find the number of lollipops Logan the Leprechaun gives away using his "lucky" method.
... more>>

The Magic of Number 6174
 Suzanne Alejandre

PreAlgebra, difficulty level 4. Have fun exploring this number magic.
... more>>

Making Money
 Alice Miller

PreAlgebra, difficulty level 3. How much money would you have by December 31, 1999, if you put 10 cents in the special "math" bank on January 1, 1999?
... more>>

Milk Madness
 Suzanne Alejandre

PreAlgebra, difficulty level 4. Help Georgina find out how much milk is left using the number pattern she describes.
... more>>

The Name Game
 Judy Ann Brown

PreAlgebra, difficulty level 3. The first student says his/her own name; the second student repeats the first student's name and adds his/her own name... How many names are said if the numbers of students in my math classes on the first day are 27, 28, 32, 31, and 30?
... more>>

Open and Shut Case
 Judy Ann Brown

PreAlgebra, difficulty level 2. Find the locker combination using mathematical clues.
... more>>

Palindrome Day
 Judy Ann Brown

PreAlgebra, difficulty level 2. How many Palindrome Days are there in a year?
... more>>

Prime Probability
 Lisa Lavelle

PreAlgebra, difficulty level 3. Given the expression n^2  5, what is the probability that the result is prime for integers between 2 and 12?
... more>>

Rampaging Reindeer
 Judy Ann Brown

PreAlgebra, difficulty level 3. How many different ways can Santa's reindeer be arranged?
... more>>

Ready for Football?
 Terry Trotter & Suzanne Alejandre

PreAlgebra, difficulty level 2. Find the final score of the game between the Eagles and the Dragons.
... more>>

Runsums, Part 1
 Ron Knott

PreAlgebra, difficulty level 3. By spotting the pattern, tell me the smallest number that is a sum of a run of one hundred consecutive numbers.
... more>>

Runsums, Part 2
 Ron Knott

PreAlgebra, difficulty level 3. Explore number patterns and report on what you have learned.
... more>>

Tiling Tables
 Suzanne Alejandre

PreAlgebra, difficulty level 3. Find a method to quickly calculate the dimensions of a rectangular table top with a given area so that it has the smallest possible perimeter.
... more>>

Treasure Hunt Math
 Judy Ann Brown

PreAlgebra, difficulty level 3. Use clues to find the mystery palindrome.
... more>>
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