turn the divisors into their reciprocals, which is done by inverting the terms, and then treat them as in multiplication. When there is a number of multiplications and divisions to be performed, it is often very convenient to throw them into a general fraction, by placing all the multipliers above a line, and all the divisors below the same, and connecting them by the sign repeated between every two. Thus, for instance, if it were required to divide the product of 12, 16, 9, 28, and 7, by the product of 49, 27, 6, and 4, we might arrange them thus, Here we leave out all the factors which are common to the two terms; and divide the product of the remaining ones above the line, by the product of those below. This is one of the most useful operations in arithmetic. It will be seen at once that this result is obtained by throwing out those factors which are common to both terms of the original fraction, and by this means the greater part of the labour of multiplying and dividing is saved. This is more a matter of convenience in practice, than of investigation of principle; but still it is so useful, that it is very desirable that every one who wishes to be an expert calculator, even in common matters, should be so well acquainted with what numbers consist of equal factors, and what do not, as to see at once how the expression may be shortened; we shall, therefore, make this the subject of the next section. 129 SECTION VII. THE FACTORS, THE DIVISORS OR MEASURES, AND THE MULTIPLES OF NUMBERS. FROM what has been shown in former sections, it will easily be understood that, regarding them merely as numbers, and without reference to their standing either for one kind of quantities or for another, there is a very great difference between numbers considered simply as numbers or answers to the question "How many?" and numbers considered as multipliers or divisors. In numbers simply considered, 1 is the standard of value; it always counts; and the symbol of no value is 0; but in a multiplier or a divisor, 1 is the standard of no value, and 0 has a very different signification. As a multiplier, 0 points out, not that there shall be no multiplication, but that there shall be no multiplicand; and 1 is really the sign of no multiplication. As a divisor, 1 is the sign that there shall be no division, and 0 is a sign that, whether the dividend be small or great, the quotient shall be infinite-shall be all possible numbers; and if we are to write it down, we may write any number whatever with equal propriety. nxo (meaning by n the greatest possible number that any one can think of) is = : 0, but the product, divided by the multiplier, gives the multipli cand; a therefore is infinitely great, and when it occurs, it is 0 usually expressed by a double 0 laid horizontally, ∞. A number which cannot be divided without remainder by any number except 1 and itself (which is no division), is called a prime number; it is an original number, or one which is not the result of any operation. K 130 PRIME AND COMPOSITE. A number which can be divided without remainder is called a composite number, because it may be said to be composed of either the quotient or the divisor, repeated as many times as the other expresses. A composite number has, then, always two divisors; and as it is composed of the product of these divisions, they are called the factors of it. Thus the factors and divisors of a number always mean the very same numbers. Still it is necessary to distinguish between them, because, when we have the factors given, whatever may be their number or value, we can in all cases find their product; but when we have a product given, we have no general means of finding what its factors may be, or whether it is a product at all. In Algebra, where the operations are expressed as well as the quantities, this difficulty is not felt; there are particular cases in which we can get the better of it in arithmetic, and no one can be expert, even as a common accountant, without being able to perceive those cases where they occur. The natural numbers taken in their order, 1, 2, 3, 4, &c., form a series or succession, beginning at 1, and increasing by the addition of 1, as a common difference; and the problem is to determine what terms of this series are prime, and what are composite. 1 is evidently a prime number, and so is 2; but we can see that 2 must be a factor of every second number after this—of 4, 6, 8, 10, 12, &c., but not of any other number. 2 is thus the smallest factor which any number can have, and the other factor corresponding to it must be half the number. Hence we are sure that half the series of the natural numbers are composite, and that no factor of a number can be greater than one-half of it. This is not much, but it is a beginning, and we may see what more we can inake of it before we proceed farther. The num bers of which 2 is not a factor, always have an odd 1 when we attempt to divide them by 2; hence we call them odd numbers, and those of which 2 is a factor are even numbers. If we divide ever so many even numbers by 2, there is not an odd 1; therefore the sum of any number of even numbers is an even number; so also is the sum of an even number of odd numbers, for the odd 1s make an even number, and when they are taken away the other numbers are all even; but if the number of odd numbers is odd, there is an odd 1, which makes the sum odd. Here we find a principle of some importance. We have seen that, if 2 divide the sum of the remainders, it will divide the sum of the numbers; and this is general, applying to any numbers and any divisor, for the single numbers are all divisible, except the remainders; and if the sum of the remainders is divisible, so must the sum of the numbers. Hence, If a number is a factor of the remainders, it is a factor of the sum; and a factor of two numbers must be a factor of their sum, their difference, their product, or any multiple of each or all of them. To return to the natural numbers;-3 is a prime number, but 4 is not; it is the second after 2, and an even number. 5 is prime, because 4 is the only composite number below it, and 5=4+1; and no factor of 4 can be a factor of 1. Hence no number and the one immediately following it in the scale can have a common factor. It is evident that every second number is divisible by 2, every third one by 3, every fourth one by 4, every fifth by 5, and so on; also that the factors of every composite number will both fall in that place of the series which answers to their numbers multiplied, and that each will then have occurred in the series as often as the other expresses. Thus 4 and 5 occur together 132 PRIME AND COMPOSITE for the first time at 20, and it is the fifth occurrence of 4, and the fourth of 5. As 1 is the measure of all numbers, whether prime or composite, it follows that all numbers whatever must fall together at some place of the series, and that there can be prime numbers only at those places where no number falls. Upon this principle we could, by mere mechanical labour, construct a table of all prime numbers, and of all divisors of composite ones, as far as we chose; and there are few better exercises for a beginner in the study of numbers, than the construction of such a table. Of course it is not necessary to go beyond half the series, as no divisor of a number can be greater than the half. The following is the arrangement as far as 20: PRIME NUMBERS AND FACTORS. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20 P. P. p. 2 p. 2 p. 2 3 2 p. 2 P. 2 3 2 P. 2 P. 2 From this example it will be seen that the factors always occur in pairs, and that if there are more than one pair, two of the divisors are composite numbers; and that when there is a 2 among the factors, there is always another factor equal to half the number. If the places of the prime numbers, which are marked by the letter P, are examined, it will be found that they are either immediately before or immediately after 2 or 3, or some number divisible both by 2 and by 3, that is, by 6. Therefore we have this general principle: —every prime number, either with 1 added to it, or 1 subtracted from it, must be divisible by 6; but |