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must also be ; for no whole number can be equal to the sum of a whole number, and a fraction. Hence the difference of the two numbers is measured by any measure of the two.

The same Prop. exhibited algebraically :- Let N and M be the two numbers, m a common measure of them: let M = m X a, N=mX b: then

M +N=m X a + m X b = m X (a + b)

.. m is contained in M + N, (a + b) times. Again, if M be greater than N, then

M = (M — N) +N
.: M: m =(M-N) m + Nm

a = (M — N);m to
.. a - b = (M - N); M
hence m is contained in M -- N, (a - b) times.

=

or

Prop. 21.-- To prove that the G.C.M. of several numbers

is the product of all the common prime factors. Since all the factors of any measure of a number are factors of the number itself, therefore all the factors of every common measure, and of the G.C.M. are factors of all the numbers. Hence all the prime factors of the G.C.M. are prime factors of all the numbers. Again since every factor, or the product of any number of co-factors, of a number is a measure of it, therefore the product all the common prime factors is a common measure of all the numbers. It is also the G.C.M. For all the factors of the G.C.M. are among these factors, therefore the G.C.M. cannot contain more than these factors, and evidently it cannot contain less than all of them, for if it could there would be a common measure greater than the G.C.M. viz. the product of all the prime factors, which is impossible. Hence the G.C.M. is the product of all the prime factors common to all the numbers.

Cor. 1. Hence every common measure of several numbers is a measure of the G.C.M. For all the prime factors of every common measure are prime factors of all the numbers ; but the G.C.M. contains all these prime factors, therefore the G.C.M. is a multiple of the product of any number of them; or the product of any nnmber of thein, i.e. any common ineasure of the numbers, is a measure of the G.C.M.

Also every nieasure of the G.C.M. being the product of some of the common primefactors of the numbers, is a common measure of them.

Cor. 2. If any of the numbers be measured by any of the others, they may be omitted in the formation of the G.C.M. For evidently no more common factors can be obtained from the multiple than from the measure; and any factor which is common to the measures, and the others, is common also to the multiples.

Prop. 22.- To prove and explain the Rule for finding the

G.C.M. of two numbers, when their prime factors are

not easily obtainable. Since the G.C.M. cannot be greater than the less of the two numbers, and may be equal to it, therefore it is first ascertained whether the less be a measure of the greater, which is done by dividing the greater by the less. If there be no remainder, the G.C.M. has been found to be equal to the less. But if there be a remainder, it is considered as follows:-Every common measure of divisor and dividend is a measure of the remainder, and therefore a common measure of divisor and remainder. For every common measure of divisor and dividend measures also the product of quotient and divisor (Prop. 19), and the dividend, and therefore measures the difference of these, or the remainder (Prop. 20); and hence is a common measure of divisor and remainder.

Again, every common measure of divisor and remainder is a measure of the dividend, and therefore a common measure of the divisor and dividend. For every common measure of divisor and remainder measures the prodnct of quotient and divisor (Prop. 19), and the remainder, and therefore measures the sum of these, or the dividend (Prop. 20), and hence is a common measure of divisor and dividend.

Hence, every common measure of divisor and dividend being a common measure of divisor and remainder, and vice versa, it follows that the G.C.M. of divisor and dividend is a common measure of the divisor and remainder, and cannot be greater than their G.C.M., else the divisor and remainder would be measured by a number greater than their G.C.M. which is absurd: nor can it be less, else there would be a common measure greater than the greatest, viz. the G.C.M. of divisor and remainder. If then we find the G.C.M. of the less and remainder, we shall have found the G.C.M. of the two given numbers. The first step in this investigation is the same as in the first, viz. to ascertain by division whether the less be the measure of the greater. If it be, the G.C.M. is found, but if not, the same considerations, as before, will shew that the G.C.M. is the same as that of the 1st and 2nd remainders, to find which has now to be tried in the same manner as the others.

From this it is concluded that if the greater of the two numbers be divided by the less, and the 1st divisor by the 1st remainder, the 1st remainder by the 2nd remainder, and so on, each remainder in its turn becoming a divisor of the previous divisor, and the division being continued till there is no remainder, then the G.C.M. of the two numbers will be that of the 1st divisor and Ist remainder, or that of 1st and 2nd remainders, or that of 2nd and 3rd, &c. or that of the last but one, and the

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last. But the G.C.M. of the last but one and the last remainders is the last remainder, which is also the last divisor. Therefore the process being as above, the last divisor will be the G.C.M.

The same Prop. exhibited algebraically:-Let a, b, be the two numbers, of which suppose b the less ; divide a by b, let the quotient be p; if there be no remainder, evidently b is the G.C.M.; but if there be a remainder, let it be c, so that a = pb + c;c=a pb.

Now every common measure of a and b measures a and pb, and ..a-pb, or c. Again, every common measure of b and c measures pb and C

and .. pb +c, or a. Hence every common measure of a and 6 is a common measure of b and c; and vice versa. Therefore the G.C.M. of a and b is a common measure of b and c, and cannot be greater than the G C.M. of b and c; nor can it be less than this, for if it were, then the G.C.M. of b and c being a common measure of a and b, a and b would have a common measure greater than the G.C.M., which is absurd. Hence the G.C.M. of a and 6 is equal to that of b and c. Let it then be tried to find the G.C.M. of b and c in the same manner as that of a and b. Divide b by c, let the

, quotient be q, the remainder d; then it may be shewn as before. that the G.C.M. of b and c is equal to that of c and d. Let then the division be continued till there is no remainder, by dividing each divisor by the remainder from each division, and it will appear by reasoning from step to step that the G.C.M. of a and 6 is equal to that of the last and last but one remainders, or to the last remainder or divisor.

Prop. 23.- To shew that the G. C. M. of several numbers

may be obtained by finding first the G. C. M. of two, then of this and a third, next of this last and a fourth, and so on; the last so obtained being the G. C. M. of the whole.

Every common measure of two numbers is a measure of their G.C.M. therefore every common measure of three is a common measure of the G.C.M. of the first two, and of the third. Again, every measure of the G.C.M. of two numbers is a commou measure of the two, therefore every common measure of the G.C.M. of two, and of a third, is a common measure of the three. Hence it appears that the G.C.M. of three numbers is the G.C.M. of that of the first two, and of the third. Similarly it may be shewn that the G.C.M. of several numbers is the G.C.M. of that of all but one, and of that one. Whence the truth of the Prop.

The same Prop. otherwise :-The G.C.M. of two numbers contains all the prime factors common to the two, and no other. Hence the G.C.M. of this and a third number contains all the prime factors common to the three, and no other; and therefore is the G.C.M. of the three. In the same way it appears that the G.C.M. of this, and a fourth, is the G.C.M. of the four; and so on, how many soever numbers there may be.

Prop. 24.- To prove the Rule for finding the L.C.M. of

several numbers.

Every number is a measure of its multiple, therefore all its prime factors are factors of the multiple. Hence all the prime factors of several numbers are factors of their common multiple ; and conversely a common multiple of several numbers contains all their prime factors, as its factors. This therefore is the only condition of a number being a common multiple of several others, viz. that it contain all their prime factors, as its factors. Now evidently, of all the common multiples which may be formed subject to this condition, that will be the least, in which the condition is only just satisfied, in which therefore no prime factor is introduced any oftener than is absolutely necessary to satisfy the condition, and in which there is no other factor than the factors of the numbers. Therefore if all the prime factors were different, the L.C.M. would be formed by multiplying all together; but if any of the same factors appear once in two or more of them, it will be sufficient to introduce it once only as a factor, into the L. C. M.; if it appear twice, three times, &c. in one, and a less number of times in others, it must be introduced, as a factor, into the L.C.M. the number of times it appears in the first, aud no oftener; for, if it were introduced less often, all the factors of the first would not appear in the L.C.M., and if it were introduced oftener, there would be more factors than are necessary. All the different factors must be introduced as before. Hence the Rule for finding the L.C.M. of several numbers evidently is :-“Resolve them into their prime factors; write down all the different factors which appear, repeating each the greatest number of times, which it appears in the same number, and multiply all together.”

Cor. 1. Hence the L.C.M. of two numbers may be found by dividing their product by their G.C.M., or by multiplying one of them by the quotient of the other divided by their G.C.M. For, the G.C.M. containing all the common prime factors, these are repeated twice in the product of the numbers, whereas in the L.C.M. they are only required to appear once. Therefore the product of the numbers is equal to the L.C.M. multiplied by the G.C.M.; or the L.C.M. is equal to the product divided by the G.C.M. And this quotient may be obtained by dividing one number and multiplying by the other.

Cor. 2. Every common multiple of several numbers is a multiple of the L.C.M. and conversely. For every common multiple must contain all the prime factors of the numbers, and the L.C.M. contains no other factors but these; therefore any other common inultiple can only be formed by introducing other factors, by which the multiple becomes a multiple of the L.C.M. The converse is evident.

Cor. 3. In forming the L.C.M. any numbers which measure any others may be omitted. For evidently if we introduce the factors of the multiples, we introduce also the factors of their measures, so that we shall not find any more factors to introduce, if we consider the measures, than if we omit them.

Prop. 25.— To shew that the L.C.M. of three or more

numbers may be obtained, by finding first the L.C.M. of two, then of this and a third; next of this last and a fourth, and so on, the L.C.M, last found being that re

quired. The L.C.M. of the first two contains all the prime factors of the first two, and no others; and the L.C.M. of three must contain all the prime factors of the three, and no others, and may therefore be found by multiplying the L.C.M. previously found, by the prime factors of the third, which are not already factors of it, and which may be obtained by dividing the third by the G.C.M. of it and the former L.C.M. That is the L.C.M. of three numbers is the L.C.M. of that of the first two and of the third. Similar reasoning will apply to four or more numbers.

Prop. 26.- To explain the necessity, method, and meaning

of the system of Fractional Numeration and Notation. In order to render any magnitude a subject for Arithmetical calculation, it is necessary to denote it by a number or numbers. The general method of doing this is, by comparing it with some fixed magnitude of the same kind, called an unit, so as to discover how many of these units are contained in it: the magnitude is then properly denoted by the number of units, because, knowing the exact magnitude of the unit, we are able to form an idea of another magnitude of the same kind, if we know the number of units, which it contains.

If every magnitude contained some one of the units in ordinary use an exact number of times, all might be denoted by ordinary numbers. But it is evident that such is not the case; for there may evidently be magnitudes less than the least of the units employed in estimating magnitudes of the same kind. And such magnitudes cannot be avoided in Arithmetical questions; therefore it is necessary to denote them in some manner by numbers.

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