Let me offer up an alternative to the decimal system. One that showcases the prime numbers.
A base-six, or senary, number system will have certain advantages to a base-ten, or decimal, number system.
Counting on your fingers becomes more efficient. It's true, we have ten fingers total, but more importantly, we have two hands of 5 fingers each. And just like we use digits 0-9 for the decimal system, you can use 0-5 for a senary system. Between our two hands, we hold 2 senary place values, where we can easily count up to 35.
Multiplitcation tables become easier. With less digits, a senary multiplication table would be easier to memorize. It also eliminates the more tedious properties of some decimal multiplication patterns. In the deciaml system, multiples of 9 & 11 are special and easy to remember, as are multiples of 2 & 8, as well as 5. But, the patterns of multiples of 3 & 7 and 4 & 6 are more complicated, and are eliminated in a senary system.
The reason a six-based number system is so compelling, is that it is composed of the first two prime numbers, 2 & 3. The decimal system skips 3 in favor of 5, making division by 3 cumbersome.
Upon inspection, we see that all the remaning primes land in the 1 & 5 spots, keeping them organized and revealing new properties, like polarity. They are either plus or minus one relative to some multiple of six.
Magnitude is another property we can assign these primes. The multiple of six that a prime resides by, is its magnitude. So, we can say that both 17 & 19 have a magnitude of 3.
To further expand upon a senary number system, rotation can aid in viewing the integers. Like a clock with only six stops, we can reference each position as relative to the multiple of six it is nearest to. 1 and 2 remain unchanged, and are considered positive. 5 becomes -1, and 4 becomes -2. Three can either be +3 or -3, whichever is most convenient.
Now, all primes greater than 3, lie on positions -1 and +1, and their polarity becomes clear. Powers of 2 alternate between positions -2 and +2. While, powers of 3, always land on +/-3.
Finding the prime factors of integers have certain tricks. For 2 and 3, a shortcut to division can be multiplication. Some multiple of 3 can be divided by 3, by multipling by 2, its senary compliment. Likewise, any multiple of 2 can be divided by 2, by multipling by 3, its senary compliment.
Dividing by primes greater than 3, have a neat trick that can only be gleaned by viewing the integers in a senary fashion. Assuming you have already factored out all 2s and 3s, you may be left with some integer ending with -1 or +1. 35, for example, is 6x6-1. Its magnitude is 6 and its polarity is negative. Looking to either side of its magnitude, we find 5 and 7, which, in fact, are its factors. Its polatiry tells us which way to look.