Let $P\left(n\right)$ and $S\left(n\right)$ denote the product and the sum, respectively, of the digits of the integer $n.$ For example, $P\left(23\right)=6$ and $S\left(23\right)=5.$ Suppose $N$ is a two-digit number such that $N=P\left(N\right)+S\left(N\right).$ What is the units' digit of $N$?

In the magic square shown, the sums of the numbers in each row, column, and diagonal are the same. Five of these numbers are represented by $v, w, x, y,$ and $z.$ Find $y+z$

A positive integer $N$ is a palindrome if the integer obtained by reversing the sequence of digits of $N$ is equal to $N$. The year $1991$ is the only year in the current century with the following two properties:

$\left(\mathrm{a}\right)$ It is a palindrome.

$\left(\mathrm{b}\right)$ It factors as a product of a $2$-digit prime palindrome and a $3$-digit prime palindrome. How many years in the millennium between $1000$ and $2000$ $($including the year $1991$$)$ have properties $\left(\mathrm{a}\right)$ and $\left(\mathrm{b}\right)$?