17.7. Question 10

Try these first

1. 1.

   What is the pigeonhole principle?

   Hint

   Think about distributing items.

2. 2.

   Provide an example of a number composed only of the digit 1 and is divisible by 2.

   Hint

   Think of how you can represent 2 with 1s.

3. 3.

   What does the pigeonhole principle guarantee?

   • At least one item must be duplicated
   • No items will be duplicated
   • All items will be distinct

   Hint

   Think about how to distribute items among limited spaces.

4. 4.

   What is the outcome of x_i - x_j if both yield the same remainder?

   • True
   • False

   Hint

   Think about the structure formed by subtracting two similar numbers.

5. 5.

   Using the pigeonhole principle, prove that among any 10 integers chosen between 1 and 10, at least two must share a common divisor greater than 1.

   Hint

   Consider how many integers share divisors.

6. 6.

   Develop a short proof that for any integer k, a number with k digits can be formed using just 0s and 1s to showcase some multiples.

   Hint

   Consider the number of digits and how they fit remainders.