Editing Senary (section)

==Mathematical properties==
{| class="wikitable floatright" style="text-align:center"
|+ Senary [[multiplication table]]
|-
! × || 1 || 2 || 3 || 4 || 5 || 10
|-
! 1
| 1 || 2 || 3 || 4 || 5 || 10
|-
! 2
| 2 || 4 || 10 || 12 || 14 || 20
|-
! 3
| 3 || 10 || 13 || 20 || 23 || 30
|-
! 4
| 4 || 12 || 20 || 24 || 32 || 40
|-
! 5
| 5 || 14 || 23 || 32 || 41 || 50
|-
!10
| 10 || 20 || 30 || 40 || 50 || 100

|}

When expressed in senary, all [[prime number]]s other than 2 and 3 have 1 or 5 as the final digit. In senary, the prime numbers are written:

:2, 3, 5, 11, 15, 21, 25, 31, 35, 45, 51, 101, 105, 111, 115, 125, 135, 141, 151, 155, 201, 211, 215, 225, 241, 245, 251, 255, 301, 305, 331, 335, 345, 351, 405, 411, 421, 431, 435, 445, 455, 501, 515, 521, 525, 531, 551, ... {{OEIS|id=A004680}}

That is, for every prime number ''p'' greater than 3, one has the [[modular arithmetic]] relations that either ''p'' ≡ 1 or 5 (mod 6) (that is, 6 divides either ''p''&nbsp;&minus;&nbsp;1 or ''p''&nbsp;&minus;&nbsp;5); the final digit is a 1 or a 5. This is proved by contradiction.

For any integer ''n'':
* If ''n'' ≡ 0 (mod 6), 6 | ''n''
* If ''n'' ≡ 2 (mod 6), 2 | ''n''
* If ''n'' ≡ 3 (mod 6), 3 | ''n''
* If ''n'' ≡ 4 (mod 6), 2 | ''n''

Additionally, since the smallest four primes (2, 3, 5, 7) are either divisors or neighbors of 6, senary has simple [[divisibility test]]s for many numbers.

Furthermore, all even [[perfect number]]s besides 6 have 44 as the final two digits when expressed in senary, which is proven by the fact that every even perfect number is of the form {{nowrap|2<sup>''p'' – 1</sup>(2<sup>''p''</sup> – 1)}}, where {{nowrap|2<sup>''p''</sup> − 1}} is prime.

Senary is also the largest number base ''r'' that has no [[totative]]s other than 1 and ''r''&nbsp;−&nbsp;1, making its multiplication table highly regular for its size, minimizing the amount of effort required to memorize its table. This property maximizes the probability that the result of an integer multiplication will end in zero, given that neither of its factors do.

If a number is divisible by 2, then the final digit of that number, when expressed in senary, is 0, 2, or 4.
If a number is divisible by 3, then the final digit of that number in senary is 0 or 3.
A number is divisible by 4 if its penultimate digit is odd and its final digit is 2, or its penultimate digit is even and its final digit is 0 or 4.
A number is divisible by 5 if the sum of its senary digits is divisible by 5 (the equivalent of [[casting out nines]] in decimal).
If a number is divisible by 6, then the final digit of that number is 0.
To determine whether a number is divisible by 7, one can sum its alternate digits and subtract those sums; if the result is divisible by 7, the number is divisible by 7, similar to the "11" divisibility test in decimal.