Problem 1

Let

\mathbb{Z}^+

be the set of positive integers. Determine all functions

f:\mathbb{Z}^+\to\mathbb{Z}^+

such that

a^2+f(a)f(b)

is divisible by

f(a)+b

for all positive integers

a

and

b

Step 5 of 6: An algebraic identity for a=p

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p^2+pf(b)=(p+b)\big(f(b)+p-b\big)-b\big(f(b)-b\big)\ \Longrightarrow\ p+b\mid b\big(f(b)-b\big)

Detailed analysis

For an odd prime $p$ and any positive integer $b$ , set $a=p$ in the original condition: $p+b\mid p^2+f(p)f(b)=p^2+pf(b)$ , using $f(p)=p$ from step 4. Expanding the right side of the displayed identity confirms it equals $p^2+pf(b)$ , so $p+b$ divides the left side; since $p+b$ obviously divides $(p+b)(f(b)+p-b)$ , it must also divide the remaining term $b(f(b)-b)$ .