The Pigeonhole Principle - Repunits

Initializing live version
Download to Desktop

Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram Player or other Wolfram Language products.

In 1834, Johann Dirichlet noted that if there are five objects in four drawers then there is a drawer with two or more objects. The Schubfachprinzip, or drawer principle, got renamed as the pigeonhole principle, and became a powerful tool in mathematical proofs.


Pick a number that ends with 1, 3, 7, or 9. Will it evenly divide a number consisting entirely of ones (a repunit)? Answer: yes. Proof: Suppose 239 was chosen. Take the remainder of 239 dividing 10, 100, 1000, ..., . The chosen number will not divide evenly into any of those 239 powers of 10, so there are 238 possible remainders, 1 to 238. By the pigeonhole principle, two remainders must be the same, for some and . As it turns out, and both give remainder 44. Subtracting, 9,999,999,000 is the result, which yields 1,111,111 when divided by 9000. When , always returns an all-1 number multiplied by 9 and some power of 10, finishing the proof. The reciprocal of the chosen number has a repeating decimal of similar length. Consider: . . ….


Contributed by: Ed Pegg Jr (March 2011)
Open content licensed under CC BY-NC-SA




Feedback (field required)
Email (field required) Name
Occupation Organization
Note: Your message & contact information may be shared with the author of any specific Demonstration for which you give feedback.