Abstract versus Real

In the article ‘of division of simple quantities’ we learnt about the process of division. We discussed the terms used in division viz. the dividend, divisor, quotient and remainder. The concept of remainder is relevant only because some numbers are not divisible by certain divisors (numbers that are not factors of our dividend).

As we have seen that some numbers are divisible by certain divisors, while others are not; it will be proper, in order to obtain a more particular knowledge of numbers, that this difference should be carefully observed, both by distinguishing the numbers that are divisible by divisors from those which are not, and by considering the remainder that is left in the division of the latter. For this purpose, let us examine the divisors; 2, 3, 4, 5, 6, 7, 8, 9, 10, &c.

First let the divisor be 2; the numbers divisible by it are, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, &c. which, it appears, increase always by two. These numbers, as far as they can be continued, are called even numbers. But there are other numbers, viz. 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, &c.

Even numbers are a special type of numbers. All the multiples of 2 are called even numbers. Every alternate integer starting from zero is an even number. All integers that are not divisible by 2 i.e., the set of all numbers preceding or succeeding even numbers by unity is called odd numbers. Of odd and even numbers, Euler writes:

Which are uniformly less or greater than the former by unity, and which cannot be divided by 2, without the remainder 1; these are called odd numbers.

The even numbers may all be comprehended in the general expression 2a; for they are all obtained by successively substituting for ‘a’ the integers 1, 2, 3, 4, 5, 6, 7, &c. And hence it follows that the odd numbers are all comprehended in the expression 2a + 1, because 2a + 1 is greater by unity than the even number 2a.

Divisibility by 2 is the way to categorise whole numbers as even and odd. In the same manner let’s categorise whole numbers on the basis of divisibility by 3.

In the second place, let the number 3 be the divisor; the numbers divisible by it are,

3, 6, 9, 12, 15, 18, 21, 24, 27, 30, and so on;

Which numbers may be represented by the expression 3a; for 3a, divided by 3, gives the quotient ‘a’ without a remainder. All other numbers which we would divide by 3, will give 1 or 2 for a remainder, and are consequently of two kinds. Those which after the division leave the remainder 1, are,

1, 4, 7, 10, 13, 16, 19, &c.

and are contained in the expression 3a + 1 ; but the other kind, where the numbers give the remainder 2, are,

2, 5, 8, 11, 14, 17, 20, &c.

Which may be generally represented by 3a + 2 ; so that all numbers may be expressed either by 3a, or by 3a -1, or by 3a + 2.

Similarly we can categorise the set of all whole numbers on the basis of their divisibility by four.

Let us now suppose that 4 is the divisor under consideration; then the numbers which it divides are,

4, 8, 12, 16, 20, 24, &c.

which increase uniformly by 4, and are comprehended in the expression 4a. All other numbers, that is, those which are not divisible by 4, may either leave the remainder 1, or be greater than the former by 1; as,

1, 5, 9, 13, 17, 21, 25, &c.

and consequently may be comprehended in the expression 4a + 1: or they may give the remainder 2; as,

2, 6, 10, 14, 18, 22, 26, &c.

and be expressed by 4a + 2; or, lastly, they may give the remainder 3; as,

3,7, 11, 15,19,23,27, &c.

and may then be represented by the expression 4a + 3.

All possible integer numbers are contained therefore in one or other of these four expressions:

4a, 4a + 1, 4a + 2, 4a + 3.

Let’s divide the set of all whole numbers on the basis of divisibility by 5 next;

It is also nearly the same when the divisor is 5; for all numbers which can be divided by it are comprehended in the expression 5a, and those which cannot be divided by 5, are reducible to one of the following expressions:

5a + 1, 5a + 2, 5a + 3, 5a + 4;

In the same manner we may continue, and consider any greater divisor. It is here proper to recollect what has been already said on the resolution of numbers into their simple factors; for every number, among the factors of which is found 2, or 3, or 4, or 5, or 7, or any other number, will be divisible by those numbers. For example; 60 being equal to 2 x 2 x 3 x 5, it is evident that 60 is divisible by 2, and by 3, and by 5.

Finding whether a number is divisible by another is sometimes of crucial importance. For large numbers, performing the process of division to judge divisibility will be a painful process. Thus, it’s important to understand the notion of divisibility rules:- A short hand way of determining whether a number is divisible by a fixed integer, without actually performing the operation of division and by simply examining the digits of the number

There are some numbers which it is easy to perceive whether they are divisors of a given number or not.

Divisibility by 2

A given number is divisible by 2, if the last digit is even; it is divisible by 4, if the two last digits are divisible by 4 ; it is divisible by 8, if the three last digits are divisible by 8 ; and, in general, it is divisible by 2″, if the n last digits are divisible by 2″.

Divisibility by 3

A number is divisible by 3, if the sum of the digits is divisible by 3 ; it may be divided by 6, if, beside this, the last digit is even ; it is divisible by 9, if the sum of the digits may be divided by 9.

Divisibility by 5

Every number that has the last digit O or 5, is divisible by 5.

Divisibility by 11

A number is divisible by 11, when the sum of the first, third, fifths ,&c. digits is equal to the sum of the second, fourth, sixth, &c. digits.

Divisibility rules

Farther, as the general expression abcd is not only divisible by a, and b, and c, and d, but also by ab, ac, ad, bc, bd, cd, and by abc, abd, acd, bcd, and lastly by abcd, that is to say, its own value; it follows that 60, or 2 x 2 x 3 x 5, may be divided not only by these simple numbers, but also by those which are composed of any two of them; that is to say, by 4, 6, 10, 15: and also by those which are composed of any three of its simple factors; that is to say, by 12, 20, 30, and lastly also, by 60 itself.

Factorising a number (breaking a number into its factors) is very useful to figure out divisibility by a number. This is because the factors of a number (dividend) compose the number; and when taken, separately, or in combination with other factors, will always divide our original number (dividend). Of the relationship between divisibility rules and factors, Euler writes:

When, therefore, we have represented any number assumed at pleasure, by its simple factors, it will be very easy to exhibit all the numbers by which it is divisible. For we have only, first, to take the simple factors one by one, and then to multiply them together two by two, It would be easy to explain the reason of these rules, and to extend them to the products of the divisors which we have just now considered. Rules might be devised likewise for some other numbers, but the application of them would in general be longer than an actual trial of the division.

It must here be particularly observed, that every number is divisible by 1; and also, that every number is divisible by itself; so that every number has at least two factors, or divisors, the number itself, and unity: but every number which has no other divisor than these two, belongs to the class of numbers, which we have before called simple, or prime numbers. Except these simple numbers, all other numbers have, beside unity and themselves, other divisors, as may be seen from the following Table, in which are placed under each number all its divisors.


Lastly, it ought to be observed that 0, or nothing may be considered as the number which has the property of being divisible by all possible numbers; because by whatever number ‘a’ we divide 0, the quotient is always 0; for it must be remarked, that the multiplication of any number by nothing produces nothing, and therefore 0 times a, or 0a, is 0.

Additional information * For example, I say that the number 53704689213 is divisible by 7, because I find that the sum of the digits of the number64004245433 is divisible by 7; and this second number is formed, according to a very simple rule, from the remainders found after dividing the component parts of the former number by 7.

Thus, 53704689213 = 50000000000 + 3000000000 + 700000000 + 0 + 4000000 + 600000 + 80000 + 9000 + 200 + 10 + 3; which being, each of them, divided by 7, will leave the remainders 6, 4, 0, 0, 4, 2, 4, 5, 4, 3, 3′, the number here given. Bernoulli.

If a, b, c, d, e, &c. be the digits composing any number, the number itself may be expressed universally thus; a +10b 102c + 103d +104c, &c. to 10nZ; where it is easy to perceive that, if each of the terms a, 10b, 10Rc, &c. be divisible by n, the number itself a + 10b + 102c, &c. will also be divisible by n.

And, if &c. leave the remainders p, q, r, &c. it is obvious, that a + 10b +102c, &c. will be divisible by n, when p + q + r, is divisible by n; which renders the principle of the rule sufficiently clear.
The reader is indebted to that excellent mathematician, the late Professor Bonnycastle, for this satisfactory illustration of M. Bernoulli’s note.

A similar Table for all the divisors of the natural numbers, from 1 to 10000 was published at Leyden in 1767, by M. Henry Anjema. We have likewise another table of divisors, which goes as far as 100000, but it gives only the least divisor of each number. It is to be found in Harris’s Lexicon Technetium, the Encyclopedia, and in M. Lambert’s Recueil, which we have quoted in the note to p.11. In this last work, it is continued as far as 102000. F.T

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