Is it really a comcoin? Very interesting background

Comcoins are a currency like any other. The residents of Multibaseania (an economically robust system of city-states with very few residents in a galaxy far, far away) use Comcoins to conduct transactions between themselves. Comcoins are represented by unique codes on special not-paper slips, and when you pay you give the vendor your slip.

However, just like any currency, there are those villainous types that try to take advantage of the system and inflate the currency because they are bored.

To combat this unquestionably illegal behavior, the Multibaseanian governments came together and devised a way to prove that the money is legitimate. The top idea from the think tank was: whenever a Comcoin is given to a vendor, the vendor runs a program that checks if it is legitimate. How does one know if a Comcoin is legitimate?


  A legitimate Comcoin is one where:
  
  When the Comcoin's unique code is converted into the bases 2-10, inclusive, that number is composite (i.e. not prime). Also, all of the unique code's digits in base 10 are either 1 or 0, and its length in base 10 is between 1 and 5 inclusive.


Very important example: in base 5, 111 is 421. Even though 111 is not prime, 421 is, so the Comcoin is invalid.

The challenge

Write a function or program that takes a Comcoin's code (a number), which is in base 10 and is an integer. Then, print or return a truthy or falsy value depending on whether that Comcoin is legitimate (criteria above), respectively.

I/O examples

Input     Output

101         False // because it is prime in base 2
1010        True
10101       True
1010111     False // because it is longer than 5 digits
123         False // because it does not contain *only* 0's and 1's
10111       False // because it is prime in base 6


This is code golf, so shortest code in bytes wins!

Edit

A Comcoin is not a number that is not prime, but is a number that is composite. 

Inspiration from qualification round of Google CodeJam 2016.
Python 2, 62 56 49 bytes
lambda n:max(`n`)&lt;&quot;2&quot;and 0x5d75d750&gt;&gt;int(`n`,2)&amp;1

Credit to @Sp3000 for the max trick to ensure binary digits.