We use to call balanced a natural squarefree number consisting of many digits (in decimal representation) as the number of his distinct prime factors.
For example, 14 is balanced as it is formed by two digits and it has two prime factors: 2 and 7
Balanced numbers of single digits are all and only the prime numbers (of course), i.e.
2, 3, 5, 7.
The balanced numbers of two digits are all the not-primes with two distinct prime factors: the smallest is
2 X 5 = 10.
Apparently, increasing the number of digits, the amount of balanced numbers increases too… but it is really easy to show that there is only a finite number of them.
Let’s take the first ten primes:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29.
Let’s multiply them: we obtain the smallest balanced number of ten digits:
There are other balanced numbers of 10 digits between 6469693230 and 9999999999. We can easily get another one, for example, by deleting 29 from the list and replacing it with 31 (the 11th prime).
We now can try to build, with the same procedure, the smallest balanced number of 11 digits, by multiplying the first 11 prime numbers … and surprise: the result, 200560490130, has 12 digits! i.e. there are no balanced numbers of eleven digits. It is easily to show, since this observation, that there are no balanced numbers of eleven or more digits.