Hence the prime factors of which 16170 is composed are 2, 3, 5, 7, 7, 11; or, 16170 = 2 × 3 × 5 × 7 × 7 × 11. EXERCISE X. EXAMPLES IN FINDING THE GREATEST COMMON MEASURE, &c. (1.) Find the G.C. M. of 285 and 465. (2.) Find the (3.) Find the of 888 and 2775. (9.) Resolve all the composite numbers from 9 to 108 into their prime factors. (10.) Resolve into their prime factors 180, 420, 714, 836, 2898, 11492, 1728, 1492, 8032, 71640, 92352, 81660 Find the G. C. M. of the following numbers by resolving them into factors: (11.) 36, 60, and 108. (12.) 56, 84, 140, and 168. (13.) 5355, 6545, 17017, 36465, 91385. LEAST COMMON MULTIPLE. 6. One number is called a multiple of another when it can be divided by the latter without a remainder. Thus, a measure and a multiple are the converse of each other. If a number divides another without remainder, it is said to be a measure of it, and the latter number is said to be a multiple of the first. A common multiple of two or more numbers is a number which can be divided by each of them without a remainder. It will clearly be a composite number, of which each of the given numbers must be a factor, for it could not otherwise be divided by them. The same numbers may clearly have an infinite number of common multiples, for any one common multiple having been found, another may be obtained by multiplying it by any number. The continued product of two or more numbers will always give a common multiple of those numbers. The least common multiple of two or more numbers is the least number which can be divided by each of them without a remainder. Thus 70 is the least common multiple of 2, 5, and 35+ 7. The least common multiple of two or more numbers is evidently composed of the continued product of all the different prime factors which compose the given numbers, each one being repeated as often as the greatest number of times it occurs in any one of the numbers. For if it did not contain all the prime factors of any one of the numbers, it could not be divided by that number. On the other hand, if any prime factor were employed more times than it is repeated in any one of the given numbers, it would not be the least common multiple. (Least common multiple is sometimes written L.C.M.) 8. EXAMPLE. Find the L. C. M. of 12, 126, and 735. These are respectively equal to 2 X 2 X 3, 2 X 3 X 3 X 7, 3 X 5 X 7 X 7. Now 2, 3, 5, 7 are all the different prime factors which occur in any of the numbers; and the greatest number of times which 2 occurs is twice-viz., in the first; the greatest number which 3 occurs is twiceviz., in the second; 5 only occurs once-viz., in the third; and the greatest. number of times which 7 occurs is twice-viz., in the third. Hence the L. C. M. required will be 2 X 2 X 3X3 X5 X 7 X 7; that is, 8820. 9. The process, then, of finding the least common multiple of two or more numbers is reduced to that of splitting up the numbers into their prime factors. This may be effected, however, by a more convenient method of arrangement than splitting each number separately into factors would be, for which we give the following Rule for finding the least common multiple of two or more numbers. Write down the numbers in a straight line apart from each other. Divide by the least number which is a measure of two or more of them, and set down the quotients and the undivided numbers in a line below. Take again the least number which is a measure of two or more of these numbers last set down, and perform the same operation as before. Continue it until there are no two numbers which can be divided by any number greater than unity. The continued product of all the divisors, and the numbers set down in the last line, will be the least common multiple required. 10. EXAMPLE.—To find the L. C. M. of 12, 42, 72, and 84. The process will be sufficiently understood from the working given in the margin. 2) 12, 42, 72, 84 2) 6, 21, 36, 42 3) 3, 21, 18, 21 Hence the L. C. M. is 6 x 7 x 3 7) I, 7, 6, 7 I, I, 6, I 2; that is, 504. This method of arrangement evidently gives the greatest number of times which each prime factor occurs in any one of the given numbers. Thus 2 occurs 3 times in 72, 3 occurs twice in 72, and 7 occurs only once-viz., in 42 and 84. EXERCISE XI. EXAMPLES IN THE LEAST COMMON MULTIPLE. Find the least common multiple of-- (1.) 15 and 45. (2.) 63 and 18. (3.) 6, 9, and 15. (4.) 8, 16, 18, and 24. (18.) 1, 2, 3, 4, 5, 6, 7, 8, and 9. (14.) 657, 350, 876, 1095, 2190, and 5795. CHAPTER VI. FRACTIONS. 1. WHEN a number or thing is divided into two equal parts, each of these parts is called one half; if the number or thing be divided into three equal parts, each is called one third; if it is divided into four equal parts, each of the parts is called one-fourth, or one quarter; and so universally when a number or thing is divided into any number of equal parts, the parts take their name from the number of parts into which the thing or number is divided. One of these parts, or a collection containing any number of them, is called a fraction of the original number or thing. Thus, if a straight line be divided into seven parts, each part is one-seventh of the line, and any number of the parts-as, for instance, five of them, i.e., fivesevenths of the whole-is a fraction of the whole line. The number of parts into which the unit or whole is divided is called the denominator, because it indicates or denominates the number of parts into which the whole is divided. The particular number of these parts taken to form any fraction of the whole is called the numerator, because it expresses the number of parts taken. Thus in the case given above, 7 is the denominator, because the line is divided into seven parts, and 5 is the numerator of the fraction. Fractions are expressed by writing the numerator above the denominator, and drawing a line between them. Thus the above fraction would be written, one half would be written, eight-ninths, and so on. 2. A proper fraction is one whose numerator is less than its denominator, as, 4, 3. 2 An improper fraction is one whose numerator is not less than its denominator, as §, 1, &c. A mixed number consists of a whole number and a fraction expressed together; for example, 3 and 4. This is generally written thus, 3; similarly, 4, 7, &c. Fractions in which the denominators are 10, or any power of 10 (Chap. IV., Art. 5), are called Decimal Fractions, or Decimals. All other fractions are called Vulgar Fractions. |