Dr. Timo de Wolff
Institute of Mathematics
www.math.tamu.edu/~dewolff/Fall14/math302.html
MATH 302 – Discrete Mathematics – Section 501
Homework 10
Fall 2014
Due: Friday, December 5th, 2014, 9:10 a.m.
When you hand in your homework, do not forget to add your name and your UIN.
Exercise 1. A group of 11 students from Texas A&M visits Europe. After their arrival
in Frankfurt they split up for the first three days. 4 visit the River Rhine and its castles,
2 go to Heidelberg and the Schwarzwald and 5 go to Hamburg. The 5 in Hamburg cannot
agree what to do on the first day after arrival – visit the harbor and the Speicherstadt or
do a mudflat hike in the nearby Wadden Sea. So, they split up: 3 go for the harbor and
2 for the hike. How many possibilities are there to distribute the students on the different
activities, if we do not know, which student takes which activity?
Exercise 2. Solve the recurrence relation
an = −8an−1 + 15
with initial conditions a0 = 1 and a1 = 1.
Exercise 3. Find the number of primes less than 200 using the principle of inclusionexclusion.
Hint: Notice that every natural number less than 200, which is not prime, has a prime
factor not exceeding 14. Hence, define the j-th set in the principle of inclusion-exclusion
as the set of numbers less than 200, which are divisible by the j-th prime lower or equal
than 14.
Exercise 4. Two integers a, b ∈ Z are congruent modulo n for some n ∈ N∗ if there
exists some k ∈ N such that a = b + k · n. In other words: a − b is divisible by n. Show
that being congruent modulo n is an equivalence relation on the integers.
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