Elementary arithmetic, with brief notices of its history... by Robert Potts.

way may be shewn that any number 7256 is not divisible by 9 when
the sum of the digits is not so divisible.
As 9 is 3 times 3, it follows that a number is divisible by 3 if the
sum of its digits be divisible by 3.1
Next, let the number 63547 be taken, and let it be divided into'
parts which are and which are not divisible by 11.
Here 63547 = 60000 + 3000 + 500 + 40 + 7
And 60000 = 6 x 10000 = 6 x (909 x 11 + 1) = 6 x 11 X 909 + 6
3000 3 x 1000 = 3 x (90 x 11 + 10) 3 x 11 x 91 -3
500=5 x 100 = 5 x (9 x 11 +1)= 5 x 11 x 9 + 5
40=4 x 10=4 x (11-1)=4 x 11 -4
7=7
Then 63547 =(6 x 11 X 909 + 3 x 11 x 91 + 5 X 11 X 9 + 4 X 11) +
(6-3 + 5-4 + 7)
The first part of which is obviously divisible by 11. Hence the
number 63547 will be divisible by 11 if the second part 6 - 3 + 5 - 4 + 7
be divisible by 11, that is if the difference between the sums of the
digits in the odd and even places reckoned from the unit's place be
divisible by 11.
In the same way may be shewn that a number is also divisible by
11, if the sum of the digits in the odd places be equal to the sum
in the even places, or when the difference between the sums in the
odd and even places is 0.
3. PRoP.-To find the prime numbers in order within any given lilat.
Let the natural numbers be written down in order, beginning withl
unity up to the given limit.
The number 2 being the only even prime number, all other even
numbers being multiples of 2 are not prime numbers and may b o
struck out.
The number 5 being the only prime number ending with 5, all
other numbers ending with 5 being multiples of 5 may be removed.
Also all numbers ending with one or more ciphers, being multiples
of 2 and 5.
1 The following simple properties are useful in resolving small composite numbers
into their prime factors:A number is divisible by 10 when it ends with a cipher.
A number is divisible by 5 when it ends with 0 or 5.
A number is divisible by 2 when it ends with a cipher or an even number.
A number is divisible by 4 when the number composed of the first two digits
reckoned from the unit's place is divisible by 4; and by 8 when the number corn.
posed of the first three digits is divisible by 8.
A number is divisible by 6 when it ends with an even number and is divisible
by 3.
A number is divisible by 12 when it is divisible by 4 and by 3.
The criterion for discovering when a number is divisible by 7 is not of a simpli
nature.