Question 44
A three-digit number with distinct non-zero digits (say $a b c$) is called *petaloso* if there exists an integer $n \geq 1$ such that the number $c b a \underbrace{00 \cdots 0}_{n \text{ zeros}}$ is a multiple of $a b c$. The smallest $n$ that makes this divisibility true is called the *fiore* (flower) of $a b c$.
Example. The number 132 is petaloso, since 132 divides 23100. Since $23100/132 = 175$ is an integer but $2310 / 132=17.5$ is not, the fiore of 132 is 2.
a. Let $a b c$ be a three-digit number (with distinct non-zero digits) of the form $2^{x} \cdot 3^{y} \cdot 5^{z}$ with $x, y, z$ non-negative integers and $y \leq 2$. Prove that $a b c$ is petaloso.
b. What is the largest possible fiore of a (three-digit) petaloso number?
c. Let $a b c$ be a petaloso number. Prove that $a b c$ is not divisible by 13.
Example. The number 132 is petaloso, since 132 divides 23100. Since $23100/132 = 175$ is an integer but $2310 / 132=17.5$ is not, the fiore of 132 is 2.
a. Let $a b c$ be a three-digit number (with distinct non-zero digits) of the form $2^{x} \cdot 3^{y} \cdot 5^{z}$ with $x, y, z$ non-negative integers and $y \leq 2$. Prove that $a b c$ is petaloso.
b. What is the largest possible fiore of a (three-digit) petaloso number?
c. Let $a b c$ be a petaloso number. Prove that $a b c$ is not divisible by 13.