Question 74
a. Find all two-digit positive integers $n$ with the following property: both integers obtained by deleting one of the two digits of the decimal representation of $n$ are (positive integer) divisors of $n$.
b. Let $n\gt 10$ be an integer written with $k$ decimal digits, all different from zero. Suppose that each of the integers obtained by deleting one of the $k$ digits of the decimal representation of $n$ is a (positive integer) divisor of $n$. Show that necessarily $k=2$.
Example. For $n=123$ we have $k=3$, and the integers obtained by deleting digits of $n$ are $23,13,12$.
b. Let $n\gt 10$ be an integer written with $k$ decimal digits, all different from zero. Suppose that each of the integers obtained by deleting one of the $k$ digits of the decimal representation of $n$ is a (positive integer) divisor of $n$. Show that necessarily $k=2$.
Example. For $n=123$ we have $k=3$, and the integers obtained by deleting digits of $n$ are $23,13,12$.