Problem 5

Let

f

be a function from the set of integers to the set of positive integers. Suppose that, for any two integers

m

and

n

, the difference

f(m)-f(n)

is divisible by

f(m-n)

. Prove that, for all integers

m

and

n

with

f(m)\le f(n)

, the number

f(n)

is divisible by

f(m)

Step 2 of 5: f is an even function

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In plain words

Comparing 0 to x and x to 0 each show one of f(x), f(-x) divides the other's gap to f(0); combined with the previous step both end up dividing each other.

f(x)=f(-x)

Detailed analysis

Setting $m=0,n=-x$ gives $f(x)\mid f(0)-f(-x)$ , which combined with $f(x)\mid f(0)$ shows $f(x)\mid f(-x)$ . Setting $m=0,n=x$ gives $f(-x)\mid f(0)-f(x)$ , which combined with $f(-x)\mid f(0)$ shows $f(-x)\mid f(x)$ . Two positive integers that divide each other are equal, so $f(x)=f(-x)$ for every integer $x$ .