Problem 1, 2018

For a natural number \(n\), let \(f(n)\) be the number written with the same digits taken in the opposite order (that is, read from right to left) whenever \(n\) is not divisible by \(10\); if \(10 \mid n\), set \(f(n) = 0\). For example, \(f(123) = 321\) and \(f(30) = 0\).

a) Show that \(n = 3f(n)\) forces \(27 \mid n\).

b) Show that \(n = 2f(n)\) forces \(9 \mid n\).