Let \(a\) and \(b\) be two different \(9\)-digit numbers. It is known that each one of them contains all of the digits \(1,2,...9\). Find the maximal value of \(\gcd(a,b)\).
For an odd number \(N\) denote by \(A\) the minimal positive difference between prime divisors of \(N\), denote by \(B\) the minimal positive difference between composite divisors of \(N\). Usually we have \(A<B\), but can we have \(A>B\)? (Disregard numbers such as \(15\) where one of \(A\) or \(B\) is not defined)
A natural number \(N\) is called perfect if it equals the sum of its divisors, except for \(N\) itself. Prove that if \(2^r-1\) is prime, then \((2^r-1)2^{r-1}\) is a perfect number. Are there any odd perfect numbers?