B2. Integers $a, b, c, d$, and $e$ satisfy the following three properties:
(i) $2 \leq a<b<c<d<e<100$
(ii) $\operatorname{gcd}(a, e)=1$
(iii) $a, b, c, d, e$ form a geometric sequence.
What is the value of $c$ ?
Solution: The answer is 36 .

Let $r$ be the common ratio of the geometric sequence $a, b, c, d, e$. Since $a1$. Then $a=a, b=a r, c=a r^{2}, d=a r^{3}, e=a r^{4}$. Since $a, e$ have no common factors and $a>1, r$ is not an integer. Let $x / y$ be this common ratio, where $x, y$ are positive integers and $\operatorname{gcd}(x, y)=1$. Since $r>1$ and is not an integer, $x>y>1$. Therefore, $b=a x / y, c=a x^{2} / y^{2}, d=a x^{3} / y^{3}$ and $e=a x^{4} / y^{4}$. Since $e$ is an integer and $\operatorname{gcd}(x, y)=1$, $a$ is divisible by $y^{4}$. Then $a=k y^{4}$ for some positive integer $k$. Then $a=k y^{4}, b=k x y^{3}, c=$ $k x^{2} y^{2}, d=k x^{3} y, e=k x^{4}$. Since $\operatorname{gcd}(a, e)=1, k=1$. Hence, $a=y^{4}$ and $e=x^{4}$. Since $2 \leq a<e<100$ and $3^{4}<100<4^{4}, 2 \leq y<x \leq 3$, which implies that $x=3$ and $y=2$. Then $c=k x^{2} y^{2}=1 \cdot 3^{2} \cdot 2^{2}=6^{2}=36$.