17.3. Lecture No#38

Try these first

1. 1.

   List the possible pairs of integers from 1 to 8 that sum to 9.

   Hint

   Look for pairs where one number increases and the other decreases.

2. 2.

   What is the midpoint formula for two points (x1, y1) and (x2, y2)?

   Hint

   Think about averaging the coordinates.

3. 3.

   What does the pigeonhole principle guarantee in the context of distinct points?

   • A. At least one point will be repeated
   • B. At least one pair of points will have integer midpoints
   • C. All points will have integer coordinates

   Hint

   Think about how points can be categorized.

4. 4.

   Choosing 5 integers from 1 to 8 always results in what?

   • True
   • False

   Hint

   Consider the defined pairs available.

5. 5.

   Consider the integers from 1 to 15. Prove that choosing any 8 integers guarantees a pair that sums to 16.

   Hint

   How many pairs can you make from the given integers?

6. 6.

   Construct a proof showing that for any integer n, a number constructed with n 1s is divisible by n.

   Hint

   Explore numbers like 1, 11, 111, and analyze their remainders.