three are a multiple of 1000, which is a multiple of 8. If, then, the last three figures are divisible by 3, so is the whole number. (2.) Similarly, and for precisely similar reasons, a num ber is divisible by 5, if its last digit is so ; i.e., if the number ends with 5 or ; by 25, if it ends with 25 or 75 ; and by 125, if it ends with 125, 625, 375, or 875. (3.) A number is divisible by 9 when the sum of its digits is so divisible. This is a case of the important pro perty that the division of a number by 9 produces the same remainder as the division of the sum of its digits by 9. Taking any number, e.g., 583, we see that it is made up of 58 tens and 3; that is, of 58 nines and 58 and 3. Again, 58 is 5 tens and 8 ; that is, 5 nines and 5 and 8. Thus, 583 is made up of 58 nines, 5 nines, and 5 + 8 + 3 ; that is, of nines + the sum of its digits. Therefore 583 and 5 + 8 + 3 must give the same remainder when divided by 9. A proof of multiplication, by " casting out the nines," depends on this property. If a number, made up, say, of nines and 7 over, be multiplied by another made up of nines and 5 over, their product must be nines and 35, that is, .nines and 8 over ; and unless this relation holds good, there must be an error in the multiplication. (4.) Similarly, a number is divisible by 3, if the sum of its digits is so ; for, if every number be made up of nines and the sum of its digits, it must be made of threes and the same sum. (5.) A number is divisible by 11, if the sums of its alternate digits are equal , or if they differ by a multiple of 11. Take any number whose alternate digits are equal, or differ by a multiple of 11, as 8294. This is equal to the sum of 8000, 200, 90, and 4 ; that is, of 80 times 99, 80, twice 99, 2, 90, and 4. Leaving out the multiples of 99 (as being multiples of 11), we have 80 + 90 + 2 + 4. But as 2 + 4 and a multiple of 11 give 8 + 9, therefore, if 80 + 90 + 2 + 4 be divisible by 11, so must 80 + 90 + 8 + 9 be, and vice versa. But 88 + 99 is divisible by 11 ; so, therefore, is 80 + 90 + 2 + 4, and so also 8294. If, in finding the greatest common measure by the method described, a divisor occurs containing a factor that evidently does not measure one of the numbers given, that factor may at once be omitted, since it can be no part of the common measure. Thus, in finding the greatest com mon measure of 59241 and 223014, we get 13950 as a divisor. Now, 50 divides this, i.e., 5x5x2, and neither 5 nor 2 measures the first of the given numbers. We can therefore reduce 13950 to 279 at once, whence we im mediately find 93 to be the greatest common measure. 10. To find the least common multiple of two given lon numbers, divide either of the numbers by the greatest p e common measure of the two, and multiply the other num ber by the quotient. Thus, the greatest common measure of 30 and 48 is 6. Therefore the least common multiple is 48 x 5 or 30 x 8, i.e., 240. For the product 5x6x8 is evidently a common multiple of 30 (i.e., 5x6) and 48 (i.e., 6 x 8), and since 8 and 5 are prime to each other, this product must be the least that contains both 5x6 and 6x8. To find the least common multiple of any given num bers, arrange them in a line, and strike out any of them that measure any of the others; take any number that measures all or part of the remainder, and divide all that it measures by that number, setting down the quotients and the undivided numbers in a second line ; proceed with this second line as with the first, and continue this process till a line is obtained of numbers prime to each other. The continued product of the divisors and the numbers in the last line is the least common multiple required. This de pends on the principle just demonstrated, that a common 29 factor of two or more numbers needs to be taken but oiice for the common multiple. In finding the least common multiple of 42, 45, 50, 54, 60, 63, 70, 75, and 90, for instance, we may divide in succession by 5, 7, 5, and 6. It is evident that 45 may be omitted, since it will measure every multiple of 90. The division by 5 is virtually the substitution for 50, 60, 70, 75, and 90, of the product 5x10x12x14x15x18, which is manifestly a common multiple of them. Then 14 and 18 are omitted, being contained in 42 and 54. Next we substitute for 42 and 63 the product 7x6x9; omit 6 and 9 as being contained in 54; then take 5x2x3, instead of 10 and 15 ; omit 2 and 3, as measures of 12; and, lastly, substitute 6x9x2 for 54 and 12. The pro duct of 5, 7, 5, 6, 9, 2, gives 18900 as the least common multiple. If the divisors are all prime numbers, the result must be the least common multiple ; and it is better to avoid using composite numbers as divisors, except when they measure a^ the numbers in the line. Had 10, for instance, been taken in the example as the first divisor, 75 would have remained in the second line, and the result obtained would have been five times too great. It is often found conveni ent to write the prime factors of the least common multiple, which is in the example 2x2x3x3x3x5x5x7. Fractions. 11. If unity be divided into any number of Fractions, equal parts, one or more of these parts is called a frac tion. If, for example, we divide unity into 7 equal parts, and take 5 of these, we shall obtain the fraction we speak of as five-sevenths. (This, and what follows, may be familiarly illustrated by taking any object and dividing it in the way described a straight line, for instance, thus, l l l I l l l l _ There are two kinds of fractions Vulgar Fractions, often spoken of simply as Fractions, and Decimal Frac tions or Decimals. A vulgar fraction is represented by two numbers, called the terms of the fraction, which are written, the one above and the other below a horizontal line ; thus, the fraction already mentioned is written T . The number under the line indicates the number of equal parts into which unity is divided, and is called the denominator, 1 as showing the " denomination " (see 32) of the fraction. The number above the line, indicating the number of those equal parts that the fraction consists of, is called the numerator. The most usual definition of a fraction is that which is given above. But it may be also defined or regarded as one number divided by another, the numerator being the dividend, and the denominator the divisor. Thus, the fraction T , which we have interpreted to mean 5 of the 7 parts into which unity is supposed to be divided, may also be regarded as the seventh part of 5 units. For, if each of 5 units be divided into 7 equal parts, there will be in all 35 of these parts, each of them equal to the seventh part of unity, and the seventh part of these 35 parts is of them. That is, the seventh part of 5 units is the samo as 5 seventh parts of unity, or T according to the former definition. It follows from this that, when there is a remainder after division, the quotient is completed by tlie addition of a fraction, of which the remainder is the 1 Though the word employed to express the denominator (e.g., the "sevenths" in " five-sevenths") agrees in form with the ordinal nume ral, this use of it is not ordinal. The seventh day means the seventh in order of time ; but the seventh part of a day is one, any ot>e of seven equal parts into which the day is regarded as divided. Some such distinct name as "fractional numerals" should be given to the words when used in the latter sense. The expressions " first part " and " second part " are never used in the fractional sense. For the latter "half" is used; there is no fractional corresponding to the former, ^j, however, is called the thirty-second part ; and so %, 7r* r , &c., are read nine thirty-seconds, four ticenty-firsts, &c.
II. 67