23. Two friends, Marco and Ian, are talking about their ages. Ian says, "My age is a zero of a polynomial with integer coefficients."
Having seen the polynomial $p(x)$ Ian was talking about, Marco exclaims, "You mean, you are seven years old? Oops, sorry I miscalculated! $p(7)=77$ and not zero."
"Yes, I am older than that," Ian's agreeing reply.
Then Marco mentioned a certain number, but realizes after a while that he was wrong again because the value of the polynomial at that number is 85 .
Ian sighs, "I am even older than that number."
Determine Ian's age.
23. Let $a$ be Ian's age. Then
$$
p(x)=(x-a) q(x),
$$
where $q(x)$ is a polynomial with integer coefficients.
Since $p(7)=77$, we have
$$
p(7)=(7-a) q(7)=77=7 \cdot 11 \text {. }
$$

Since $q(7)$ is an integer and $7-a<0$, we restrict
$$
a-7 \in\{1,7,11,77\} \text {. }
$$

Let $b$ be the second number mentioned by Marco. Since $p(b)=85$, we have
$$
p(b)=(b-a) q(b)=85=5 \cdot 17 .
$$

Since $q(b)$ is an integer and $b-a<0$, we restrict
$$
a-b \in\{1,5,17,85\} .
$$

Finally, we know from algebra that $b-7$ is a divisor of $p(b)-p(7)=$ $85-77=8=2^{3}$. It follows that
$$
b-7 \in\{1,2,4,8\} .
$$

Considering all the possibilities from ( $\star \star$ ) and ( $\star \star \star)$, since $a-7=$ $(a-b)+(b-7)$, we get
$$
a-7 \in\{2,3,5,6,7,9,13,18,19,21,25,86,87,89,93\} .
$$

Recalling $(\star)$, we get $a-7=7$ or $a=14$.