Math Problem Statement
Let n be a positive integer. Prove that there is a number of the form 1 · · · 10 · · · 0 (this is a number whose first digits, from the left, are 1’s and whose last digits are 0’s; numbers with a zero number of 0’s are considered as well) that is divisible by n.
Solution
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Math Problem Analysis
Mathematical Concepts
Number Theory
Divisibility
Modular Arithmetic
Formulas
a_k = \frac{10^k - 1}{9}, where a_k is a number made up of k ones
a_j - a_i = \underbrace{111\cdots1}_{(j-i) \text{ ones}} \underbrace{000\cdots0}_{i \text{ zeros}}
Theorems
Dirichlet's Box (Pigeonhole) Principle
Suitable Grade Level
Undergraduate
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