OPERATION. A COMMON DIVISOR. Art. 118. A common divisor of two or more numbers is any number that will divide them without a remainder; thus, 2 is a common divisor of 2, 4, 6, and 8. Art. 119. To find a common divisor of two or more numbers. Ex. 1. What is the common divisor of 10, 15, and 25 ? Ans. 5. We resolve each of the given numbers into two 10=5 X 2 factors, one of which is common to all of them. 15 5 X 3 In the operation 5 is the common factor, and there25: 5 x 5 fore must be a common divisor of the numbers. RULE. -Resolve each of the given numbers into two factors one of which is common to all of them, and this common factor is a ammon divisor. EXAMPLES FOR PRACTICE. 2. What is the common divisor of 3, 9, 18, 24 ? Ans. 3. 3. What is the common divisor of 4, 12, 16, 28 ? Ans. 2 or 4. Art. 120. A divisor of any factor of a number is a divisor of the number itself. Thus 3, a divisor of 9, a factor of 45, is a divisor of 45 itself. Art. 121. A common divisor of two numbers is a divisor of their sum and of their difference. Thus 4, a common divisor of 16 and 12, is a divisor of their sum, 28, and of their differ ence, 4. ART. 122. A common divisor of the remainder and the divisor is a divisor of the dividend. Thus, in a division having 12 for remainder, 36 for divisor, and 48 for dividend, 12, a common divisor of the 12 and the 36, is also a divisor of the 48. THE GREATEST COMMON DIVISOR. Art. 123. The greatest common divisor of two or more numbers is the greatest number that will divide each of them without a remainder. Thus 6 is the greatest common divisor of 12, 18, and 24. QUESTIONS. Art. 118. What is a common divisor of two or more numbers ? - Art. 119. What is the rule?- Art. 121. Of what is the common divisor of two numbers a divisor ? - Art. 122. Of what is a common divisor of the less of two numbers and of their difference a divisor ? - Art. 123. What is the greatest common divisor of two or more numbers ? FIRST OPERATION. SECOND OPERATION. ART. 124. To find the greatest common divisor of two or more numbers. Ex. 1. What is the greatest common divisor or measure of 84 and 132 ? Ans. 12. Resolving the numbers into their 84= 2 X 2 X 3 X 7 prime factors (Art. 114), thus, 84 132 2 X 2 X 3 X 11 = 2 X 2 X3 X 7, and 132=2X 2 X 2 X 3=12. 2 X3 X 11, we find the factors 2 X 2 X 3 are common to both. Since only these common factors, or the product of two or more of such factors, will exactly divide both numbers, it follows that the product of all their common prime factors must be the greatest factor that will exactly divide both of them. Therefore 2 X 2 X 3 12 is the greatest common divisor required. The same result may be obtained by a sort of trial process, ag by the second operation. It is evident, since 84 cannot 84) 132 (1 be exactly divided by a number 84 greater than itself, if it will also exactly divide 132, it will 48) 84 (1 be the greatest common divisor 48 sought. But, on trial, we find 84 will not exactly divide 132, 3 6 ) 48 (1 there being a remainder, 48. 36 Therefore 84 is not a common 12) 36 (3 divisor of the two numbers. We know a common divisor 36 of 48 and 84 will also be a divisor of 132 (Art. 122). We next try to find that divisor. It cannot be greater than 48. But 48 will not exactly divide 84, there being a remainder, 36; therefore 48 is not the greatest common divisor. Again, as the common divisor of 36 and 48 will also be a divisor of 84 (Art. 122), we try to find that divisor, knowing that it cannot be greater than 36. But 36 will not exactly divide 48, there being a remainder, 12; therefore 36 is not the greatest common divisor. As before, the common divisor of 12 and 36 will be a divisor of 48 (Art. 122) ; we make a trial to find that divisor, knowing that it cannot be greater than 12, and find 12 will exactly divide 36. Therefore 12 is the greatest common divisor required. RULE 1. - Resolve the given numbers into their prime factors. The product of all the factors common to the several numbers will be the greatest common divisor. Or, RULE 2. Divide the greater number by the less, and if there be a QUESTION. - Art. 124. What are the rules for finding the greatest common divisor of two or more numbers ? remainder divide the preceding divisor by it, and so continue dividing until nothing remains. The last divisor will be the greatest common divisor. NOTE. — When the greatest common divisor is required of more than two numbers, find it of two of them, and then of that common divisor and of one of the other numbers, and so on for all the given numbers. The last common divisor will be the greatest common divisor required. EXAMPLES FOR PRACTICE. 2. What is the greatest common divisor of 85 and 95 ? Ans. 5. 3. What is the greatest common divisor of 72 and 168 ? Ans. 24. 4. What is the greatest common divisor of 119 and 121 ? Ans. 1. 5. What is the greatest common divisor of 12, 18, 24, and 30 ? Ans. 6. 6. Having three rooms, the first 12 feet wide, the second 15 feet, and the third 18 feet, I wish to purchase a roll of the widest carpeting that will exactly fit each room without any cutting as to width. How wide must it be? Ans. 3 feet. A COMMON MULTIPLE. ART. 125. A multiple of a number is a number that can be divided by it without a remainder ; thus 6 is a multiple of 3. ART. 126. A common multiple of two or more numbers is a number that can be divided by each of them without a remainder ; thus 12 is a common multiple of 3 and 4. ART. 127. The least common multiple of two or more numbers is the least number that can be divided by each of them without a remainder ; thus 30 is the least common multiple of 10 and 15. NOTE. — A multiple of a number contains all the prime factors of that number ; and the common multiple of two or more numbers contains all the prime factors of each of the numbers. Therefore, the least common multiple of two or more numbers must be the least number that will contain all the prime factors of them, and none others. Hence it will have each prime factor taken only the greatest number of times it is found in any of the several numbers. QUESTIONS. – Art. 125. What is a multiple of a number ? - Art. 127. What is the least common multiple of a number? FIRST OPERATION. - We SECOND OPERATION. ART. 128. To find the least common multiple. Ans. 36. Resolving the numbers into their 6 2 X 3 prime factors, thus, 6=2X 3, and 9=3X3 9=3 X 3, and 12=2 X 2 X 3, 12 = 2 X 2 X 3 find their different prime factors to be 2 X 2 X 3 X 3= 36 2 and 3. The greatest number of times the 2 occurs as a factor in any of the numbers is twice, as 2 X 2 in 12; and the greatest number of times the 3 occurs in any of the numbers is also twice, as 3 x 3 in 9. Hence 2 X 2 X3 X 3 must be all the prime factors that are necessary in composing 6, 9, and 12; and, consequently, the product of these factors must be the least number that can be exactly divided by 6, 9, and 12. Therefore 2 X2 X3 X 3=36 is the least common multiple required. Another method, and one usually 3 6 9 12 preferred, is as by second operation. 2 2 3 4 Having arranged the numbers on a horizontal line, we divide by 3, a prime 1 3 2 number that will divide all of them 3 X 2 X 3 X 2= 36 without a remainder, and write the quotients in a line below. We next divide by 2, a prime number that will divide without a remainder most of them, writing down the quotients and undivided numbers as before. Then, since these numbers are prime to each other, we multiply together the divisors and the numbers on the lower line, which are all the prime factors of 6, 9, and 12, and thus obtain 36 for the least common multiple. RULE 1. Resolve the given numbers into their prime factors. The product of these factors, taking each factor the greatest number of times it occurs in any of the numbers, will be the least common multiple. Or, RULE 2. —Having arranged the numbers on a horizontal line, divide by such a prime number as will divide most of them without a res mainder, and write the quotients and undivided numbers in a line beneath. So continue to divide until no prime number greater than 1 will divide two or more of them. The product of the divisors and the numbers of the line below will be the least common multiple. Note 1.- When numbers are prime to each other, their product is their least common multiple. NOTE 2. When one or more of the given numbers are factors of any one of the other numbers the factor or factors may be cancelled. QUESTION. multiple ? Art. 128. What are the rules for finding the least common OPERATION. EXAMPLES FOR PRACTICE. 2. What is the least common multiple of 7, 14, 21, and 15? Ans. 210. 717 14 21 15 Since 7 is a factor of 14, another of the numbers, we cancel it; and since 3 is a 2 $ 15 factor of 15, we also cancel that (Note 2): thus the work is rendered shorter. 7 X 2 X15=210 3. What is the least common multiple of 3, 4, 5, 6, 7, and 8? Ans. 840. 4. What is the least number that 10, 12, 16, 20, and 24, will divide without a remainder ? Ans. 240. 5. What is the least common multiple of 9, 8, 12, 18, 24, 36, and 722 Ans. 72. 6. Five men start from the same place to go round a certain island. The first can go round it in 10 days; the second, in 12 days; the third, in 16 days; the fourth, in 18 days; the fifth, in 20 days. In what time will they all meet at the place from which they started ? Ans. 720 days. 9 XVIII. FRACTIONS. ART. 129. A FRACTION is an expression denoting one or more equal parts of a unit. The term fraction is derived from the Latin word frango, which signifies to break; from the idea that a number or thing is broken or separated into parts. Fractions are of two kinds, Common and Decimal. COMMON FRACTIONS. ART. 130. A COMMON FRACTION is expressed by two numbers one above the other, with a line between them. The number below the line is called the denominator ; and the number above, the numerator. Thus, Numerator 3 Three Denominator 5 Fifths. QUESTIONS. — Art. 129. What is a fraction ? From what is the term derived, and what does it signify? How many kinds of fractions, and what are they called ? - Art. 130. How is a common fraction expressed ? What is the number below the line called? The number above the line ? |