RULE. Divide in succession the greater number by the less, and that divisor by the last remainder, till nothing remains. The last divisor will be the great est common measure. 70. This rule depends upon the principle illustrated in (ART. 52,) viz., that any number that measures any other number will also measure their sum, their difference, and any multiple of those numbers. 1. What is the greatest common measure of 720 and 612? 612) 720 ( 1 108 ) 612 ( 5 540 72 ) 108 ( 1 72 36 ) 72 ( 2 72 The greatest number being divided by the less, and the last remainder by the last divisor, gives 36 as the greatest common measure. Since 36 measures 72, it will also measure 108, the sum of 36 and 72; it will also measure 612, a multiple of 36. EXAMPLES. 2. What is the greatest common measure of 252 and 348? 3. What is the greatest common measure of 493 and 899? 4. What is the greatest common measure of 208 and 648? Recite the rule, when the numbers are large. 5. What is the greatest common measure of 825 and 960? 6. What is the greatest common measure of 5184 and 6912? 7. What is the greatest common measure of 3242 and 7564? 8. What is the greatest common measure of 972 and 1468? 9. What is the greatest common measure of 656, 864, and 976 ? 10. What is the greatest common measure of 896, 764, and 938? 11. What is the greatest common measure of 372, 964, and 704? THE LEAST COMMON MULTIPLE. 1. The least number divisible by two or more numbers is the least common multiple of those numbers, (ART. 47.) 72. To find the least common multiple of two or more numbers, RULE. First cancel every number that will measure any other given number. Then cancel the largest factor common to the two largest numbers, and all the factors of the other numbers that are contained in the uncancelled factors. The continued product of the remaining figures will be the least common multiple. OBS. 1. The numbers should be written in order in a horizontal line, beginning with the least on the left, and ending with the largest on the right. OBS. 2. The largest common measure and the least common multiple should not be confounded. The largest common measure is the largest number that will divide two or more numbers. The least common multiple is the least number that is divisible by two or more numbers. 73. It is evident that the least common multiple of two numbers is the product of the numbers after striking What is the least common multiple? What is the rule? E out their greatest common divisor, or the largest factor common to them, as it is necessary that a number be contained in the product as a factor only once. The principle is the same for finding the least common multiple for three or more numbers. 1. What is the least common multiple of 6, 8, 12, 16, 20, 24? 6, 8, 12, 16, 20, 24. 2×5×24=240. First cancel 6, 8, 12, since they measure 24. Then cancel 4 in 20, since it is the largest factor common to 20 and 24, and write underneath the other factor, 5. Cancel 8 in 16, as it is the largest factor contained in 24. The other factors, 2×5×24-240, the least common multiple. 2. What is the least common multiple of 14, 18, 21, 27, 28, 126? Cancel Cancel 14, 18, 21, since they measure 126. also 14 in 28, the largest factor common to 28 and 126, and also 9 in 27, the largest factor common to the uncancelled factors in 2 and 126. The product of 3×2×126=756, the least common multiple. EXAMPLES. 3. What is the least common multiple of 4, 6, 8, and 10? 4. What is the least common multiple of 3, 4, 7, and 14? 5. What is the least common multiple of 5, 7, 10, and 21? 6. What is the least common multiple of 6, 9, 18, and 24? Upon what principle does the rule depend? 7. What is the least common multiple of 8, 12, 16, and 20? 8. What is the least common multiple of 10, 12, 16, 18, and 20? 9. What is the least common multiple of 2, 3, 4, 5, 6, 7, 8, 9? 10. What is the least common multiple of 18, 36, 72, and 108? 11. What is the least common multiple of 14, 21, 28, and 35? 12. What is the least common multiple of 15, 20, 35, 45, and 60? 13. What is the least common multiple of 23, 46, 69, and 92? 14. What is the least common multiple of 34, 51, 68, and 85? 15. What is the least common multiple of 37, 74, 111, and 148? 16. What is the least common multiple of 41, 82, 123, and 164? 17. What is the least common multiple of 25, 60, 75, and 100? 18. What is the least common multiple of 30, 70, 90, and 140? 19. What is the least common multiple of 65, 75, 95, and 125? 20. What is the least common multiple of 136, 144, and 284? 21. What is the least common multiple of 272, 384, and 756? 22. What is the least common multiple of 320, 450, and 640? 23. What is the least common multiple of 426, 560, and 846? 24. What is the least common multiple of 576, 646, and 736? FRACTIONS. SECTION VII. 74. A FRACTION is one or more equal parts of a unit. 75. Fractions are of two kinds, common and decimal. A common fraction is composed of two terms, one written above the other, with a line between them. Thus, 4, 3. The term below the line is called the denominator, and shows the number of equal parts. into which the unit is divided. The term above the line is called the numerator, and shows how many parts are expressed by the fraction. OBS. The term numerator is from a Latin word, signifying to number; the denominator from a Latin word, signifying to name. 76. A fraction may also be considered as the quotient resulting from division, the numerator being the dividend, and the denominator the divisor. Thus is the quotient resulting from 4 being divided by 7, and may be read one seventh of four, or four sevenths of one. In the same manner, one. is read one third of two, or two thirds of one. 77. A proper fraction is one whose numerator is less than its denominator, as 2, 3, 4. 5 7 78. An improper fraction is one whose numerator is equal to or greater than its denominator, as 4, 4, 6. 9 5 OBS. Proper and improper fractions are also called simple fractions. What are fractions? How many kinds of fractions? Of what is a fraction composed? How are fractions written? What are the terms of a fraction called? What does the term below the line show? What does the term above the line show? How may a fraction be considered? How are fractions read? What is a proper fraction? An improper fraction? A simple fraction? |