Distinct Possibilities for Renting Apartments

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1 –
Jack and Jill want to rent separate apartments on the third floor of a new building by the river.
The building has nine apartments available, numbered 301, 302,. . . , 309. The odd-numbered
apartments have a river view, and the even-numbered apartments do not. Jill will only rent an
apartment with a river view, and Jack does not care about the view. How many distinct
possibilities exist for the pair of apartments they end up renting?
Solution here >
$Let$
$$A= set of apartments of third floor$$
$$B= Set of odd-numbered apartments $$
$$C= Set of even-numbered apartments$$
$$A \{1, 2, 3, 4, 5, 6, 7, 8, 9\}$$
$$B \{1, 3, 5, 7, 9\}$$
$$C \{2, 4, 6, 8\}$$
$$=n(A)=9 n(B)= 5 n(C)= 4$$
$$= (A ∩ B)x(C)$$
$$= 5 x 4$$
$$= 20$$
2 –
$n ≥ 2$ distinguishable Hogwarts students participate in Professor Snape’s
experiment. Each student is given potion $A$, or philter $B$, or neither, or
both. We know that Harry and Hermione are the only students among the n that
are given both $A$ and $B$. In how many distinct ways could Snape have
distributed his experimental liquids?
Solution here >
$$n ≥ 2 $$
$$A U B U {∅} U AB$$
$$n(A)$$
$$n(B) $$
$$n({∅})$$
$$n(AB)$$
= $4 distinct ways$
3 –
Prove that for all integers $x$, if $x$ is odd and $x ≥ 3$, then $x^2 − 1$ is
divisible by 8.
Solution here >
If $x$ is odd, then there exists $k∈Z$, such that $x = 2k+1$. Therefore:
$$x^2−1=(2k+1)^2−1=4k^2+4k+1−1=4k^2+4k$$
$$=8×\frac{k(k+1)}{2}$$
$k(k+1)$ is even, since either $k$ or $k+1$ is even. Thus we can write it as
$2n$ where $n∈Z$, so now we obtain:
$$x^2−1=8\frac{2n}{2}=8n$$
Therefore $x^2−1$ is divisible by 8.
4 –
(a) A number $n ∈ N$ is called perfect if the sum of all of $n’s$ factors
other than $n$ itself is equal to $n$. For example, $6$ is perfect because its