Practice Question 10 (17.7) - Module No#08 - Discrete Mathematics

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6 sample questions

Question 1.
What is the pigeonhole principle?
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Think about distributing items.
Question 2.
Provide an example of a number composed only of the digit 1 and is divisible by 2.
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Think of how you can represent 2 with 1s.
3Question 3.
What does the pigeonhole principle guarantee?
- At least one item must be duplicated
- No items will be duplicated
- All items will be distinct
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Think about how to distribute items among limited spaces.
4Question 4.
What is the outcome of x_i - x_j if both yield the same remainder?
- True
- False
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Think about the structure formed by subtracting two similar numbers.
Question 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.
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Consider how many integers share divisors.
Question 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.
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Consider the number of digits and how they fit remainders.

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