WOLFRAM|DEMONSTRATIONS PROJECT

The Pigeonhole Principle - Repunits

​
pick a prime
111111111111111
31
=
3584229390681
1
31
=
0.
0
3
2
2
5
8
0
6
4
5
1
6
1
2
9
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, ...,
239
10
. 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
a
10
and
b
10
. As it turns out,
3
10
and
10
10
both give remainder 44. Subtracting, 9,999,999,000 is the result, which yields 1,111,111 when divided by 9000. When
b>a
,
b
10
-
a
10
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:
1,111,111/239=4649
.
4649×9=41841
.
1/239=.00418410041841
….