MathDB

f(xy) = f(x)f(y), f(30) = 1, for any n whose last digit is 7, f(n) = 1

Source: New Zealand NZMOC Camp Selection Problems 2013 p5

September 19, 2021

functionfunctionalalgebrafunctional equation

Problem Statement

Consider functions

f

from the whole numbers (non-negative integers) to the whole numbers that have the following properties:

\bullet

For all

x

and

y

f(xy) = f(x)f(y)

\bullet

f(30) = 1

, and

\bullet

for any

n

whose last digit is

7

f(n) = 1

. Obviously, the function whose value at

n

1

for all

n

is one such function. Are there any others? If not, why not, and if so, what are they?