About "interesting" numbers #2

We call a divisor of a positive integer interesting if it is a divisor of a number and differs from 1 and the number itself. We call a number $X$ very interesting if it has at least two interesting divisors and it is divisible by the difference between any two of its interesting divisors.

How can one find the product of all very interesting numbers?

I experimented numerically and came to the conclusion that such numbers never exceed 1000. Is it really so?

Solution 1:

Suppose $n$ is very interesting.

Then $n$ must be composite, hence we can write, $n=ab$, where $a$ is the least prime factor of $n$, and $b > 1$.

If $b \le a$, then by minimality of $a$, we must have $b=a$, but then $n=a^2$, which is not possible, since the square of a prime is not a very interesting number. Therefore $b > a$.

If $a$ is odd, then $n$ is odd, hence so is $b$.

But then $b-a$ is even, hence, since $(b-a){\,\mid\,} n$, it follows that $2{\,\mid\,}n$, contradiction.

Hence we must have $a=2$, so $n=2b$.

Then $b-2$ must divide $n$, but then, since $$n = 2b = 2(b-2) + 4$$ it follows that $(b-2){\,\mid\,}4$, hence $b \le 6$, so $n \le 12$.

For $1 \le n \le 12$, the only candidates are $6,8,10,12$, since those are the only even composite numbers which are not the square of a prime.

The number $10$ is not very interesting, since $5-2 = 3$, which is not a divisor of $10$.

It's easily verified that the numbers $6,8,12$ are very interesting, hence those are the only very interesting numbers.