1) Prove that ifx, yare arbitrary integers then

x²y+xy² is an even number.

Remark.One option is to do a proof by cases.

2. Ifais an integer then there areat most

oneintegerqandat most oneintegerrsuch that

(i) 0r<5,and (ii)a=5q+r

We then formulated a general result with an integerb >1 in place of 5. Formulate this general result withbin place of 5 again, and then give a proof of the general result, which mimics the proof with number 5 in the lecture. Write the proof carefully. There are one or two spots in the proof which require a little bit

more care than the special case withb= 5.

3. Using induction onn, that ifn5 is an integer

andAis a set withnelements, then

there exist preciselyn!/(5!.(n-5)!)subsets ofAwhich have exactly 5 elements.

Prove by induction onnthe following generalization of this result, where number

5 is replaced with some arbitraryk0:

There exist preciselyn!/(k!.(n-k)!)subsets ofAwhich have exactlykelements.

4. Using strong induction, the following statement: () Every integer larger than 1 is divisible by a prime number.

We say that an integernis aproduct of primesiff there exist some integerk >0 and prime numbersp1,...,pksuch thatn=p1p2 pk. Notice in particular that it is allowed thatk= 1, so every prime number is a product of primes in the above sense. Notice also that the primesp1, . . . , pkarenotrequired to be distinct.

Use strong induction to prove the following statement, which is a strengthening of () above:

() Every integer larger than 1 is a product of primes.