ANALYSIS OF COMPOSITE NUMBERS. 89.a. All composite numbers are composed of prime factors; that is, the ultimate or least factors into which they may be resolved, are prime numbers. Hence, OBS. All numbers are either prime numbers, or are composed of prime factors. Ex. 1. Resolve 60 into its prime factors. Suggestion.-Divide the 60 by 2, which is the least number that will divide it without a remainder. In like manner divide this quotient by 2; and the next quotient by 3. The divisors, 2, 2, 3, with 5, the last quotient, are the prime factors required. Hence, Operation. 2)60 2)30 3)15 89.6. to resolve a composite number into its prime factors, Divide the given number by the smallest number which will divide it without a remainder; then divide this quotient in the same way, and thus continue the operation till a quotient is obtained, which can be divided by no number greater than 1. The several divisors with the last quotient, will be the prime factors required. OBS. The reason of this rule may be seen from the following considerations: First, the respective divisors are prime factors; for, they are the least numbers, which will divide the given number and the successive quotients, without a remainder. (Art. 89. Def. 4. Obs.) Second, the last quotient is also a prime factor; for, it can not be exactly divided by any number except a unit and itself. (Art. 89. Def. 4.) 2. A composite number can be divided by any of its prime factors without a remainder, and by the product of any two or more of them, but by no other number Thus, the prime factors of 30 are 2, 3, and 5. Now 30 can be divided by 2, 3, and 5; also by 2×3, 2× 5, 3 × 5, and by 2×3×5; but by no other num ber. 2. Resolve 24, 26, 32, 34, 36, 39, and 44 into prime factors. 3. Resolve 46, 48, 51, 52, 58, 62, 68, 69 into prime factors. 4. Resolve 70, 72, 74, 75, 76, 78, 82, 85 into prime factors. 5. Resolve 120, 124, 136, 156, 208, 145, 225, into prime fact. 6. Resolve 256, 344, 576, 672, 796, 864, 945, into prime fact. 7. Resolve 3420,18500, 46096, and 96464 into prime factors. QUEST.-89. a. Of what are all composite numbers composed? Obs. What then is true of all numbers? 89. b. How do you resolve a composite number into its prime factors? Obs. How does it appear that the divisors and last quotient are prime factors? CANCELLATION.* 90. Cancellation is the method of contracting arithmetical operations by rejecting equal factors. OBS. 1. The term cancel, is from the Latin cancello, to cross out, or reject. 2. Cancellation is applicable to division, when the divisor and dividend have common factors, and especially to that class of examples which involve both multiplication and division. But it is applied with the greatest advantage to reduction of Compound Fractions to Simple ones, Multiplication and Division of Fractions, Simple and Compound Proportion. 90.a. We have seen that division is finding a quotient, which multiplied into the divisor will produce the dividend. (Art. 65.) If, therefore, the dividend is resolved into two such factors that one of them is the divisór, the other factor will be the quotient. Suppose 42 is to be divided by 6. Now the factors of 42 are 6 and 7, the first of which being the divisor, the other must be the quotient. Therefore, Canceling a factor of any number, divides the number by that factor. Hence, 91. When the divisor is a factor of the dividend. Cancel both the divisor and the factor of the dividend to which it is equal; and the remaining factor of the dividend will be the quotient. 1. Divide the product of 19 into 25 by 19. Suggestion.-Cancel the 19, which is common both to the divisor and dividend, and 25, the other factor, is the quotient. Operation. 19)25×19 25 Ans. OBS. 1. It should be observed, whenever a factor canceled, is equal to the number itself, the figure 1 either expressed or understood, is always left; for if any number is divided by itself, the quotient is always 1. (Art 82. Obs.) 2. When the 1 thus left, is in the multiplier or divisor or denominator of a fraction, it may be disregarded; for, multiplying or dividing a number by 1, does not alter the number. (Arts. 45, 82. Obs. 4.) But when the 1 stands in the dividend, or numerator of a fraction, it performs an important office, and must be preserved. QUEST.-90. What is Cancellation? Obs. What is meant by the term cancel? To what rules is cancellation applicable? 90.a. What is the effect of canceling a factor of any number? 91. When the divisor is a factor of the dividend, how do you proceed? *Birk's Arithmetical Collections, London, 1764. 2. Divide 85 × 31 by 85. 3. Divide 76 × 58 by 58. 4. Divide 75 x 40 by 40. Operation. Suggestion.-284x7; consequently the whole dividend is equal to 63 x 4 × 7. We therefore cancel the 7, which is a factor common both to the dividend and the divisor. The product of 63 × 4, the other factors of the dividend, is the answer required. 6. In 32 times 84, how many times 8? 7. In 35 times 95, how many times 7? 8. In 48 times 133, how many times 8? 9. In 96 times 156, how many times 12? 10. Divide 168 × 2 × 7 by 7 × 3. Suggestion. We first cancel the factor 7, Ans. 336. Operation. which is common to the divisor and divi-3)168×2×1 dend, then divide the product of 168 into 2 by 3. 3)336 11. Divide the product of 21 into 4 into 9, by the product of 3 into 7 into 2 into 3. Suggestion. The product of 3 x7 Operation. is 21; we therefore cancel the fac- $XX2×3)21×4×9 6)36 6 Ans. tors 3 and 7 in the divisor, and the 21 in the dividend. Then dividing the product of 4×9 the remaining factors of the dividend by the product of 2 × 3, the remaining factors of the divisor, we have 6 for the quotient. Hence, 92. When the divisor and dividend have common factors. Cancel the factors common to both; then divide the product of the remaining factors in the dividend by the product of those remaining in the divisor, and the quotient will be the answer. OBS. If two or more factors in the divisor taken together, aro equal to one factor in the dividend, or vice versa, cancel the single factor in the one, and those which are equal to it in the other. 12. Divide the product of 7, 9, 15, and 8 by the product of 5, 7, and 8. 13. Divide the product of 6, 3, 7, and 4 by 12 into 6. 14. Divide the product of 2, 28, and 15 by 30. 15. Divide the product of 5, 6, and 56 by 7 into 8. QUEST.-92. When the divisor and dividend have common factors, how proceed? GREATEST COMMON DIVISOR. 93. A Common Divisor is a number which will divide two or more numbers without a remainder. divisor of 4, 6, 8, 12, 16. Thus, 2 is a common 94. The Greatest Common Divisor of two or more numbers, is the greatest number which will divide each of them without a remainder. Thus, 6 is the greatest common divisor of 12, 18, and 24. OBS. 1. A common divisor is often called a common measure. 2. Numbers which have no common measure or divisor, greater than 1, are said to be incommensurable. Thus, 11 and 17 are incommensurable. 3. It will be seen that a common divisor of two or more numbers, is simply a factor which is common to those numbers, and the greatest common divisor is the greatest factor common to them. Hence, 95. To find a common divisor of two or more numbers. Resolve each number into two or more factors, one of which shall be common to all the given numbers. (Art. 56.) OBS. If the given numbers have not a common factor, they cannot have a common divisor greater than 1; consequently, they are either prime numbers, or are prime to each other. (Art. 89. Def. 5. Obs. 2.) Note. The following facts may assist the learner in finding common divisors: 1. Any number ending in 0, also any even number, as 2, 4, 6, &c., may be divided by 2. 2. Any number ending in 5 or 0, may be divided by 5. 3. Any number ending in 0, may be divided by 10. 4. When the two right hand figures are divisible by 4, the whole number may be divided by 4. 5. If the three right hand figures of any number are divisible by 8, the whole is divisible by 8. Ex. 1. Find a common divisor of 8, 10, and 12. Analysis.-8 may be resolved into the factors 2 and 4; that is, 8=2×4; 10=2×5; and 12=2×6. The factor 2 is common to each number and is therefore a common divisor of them. 2. Find a common divisor of 9, 15, 18, and 24. 3. Find a common divisor of 16, 20, and 36. 4. Find a common divisor of 35, 50, 75, and 80. 5. Find a common divisor of 148 and 184. 6. Find a common divisor of 126 and 4653. QUEST.-93. What is a common divisor? 94. What is the greatest common divisor of two or more numbers? Obs. What is a common divisor sometimes called? 95. How do you find a common divisor of two or more numbers? Operation. 7. What is the greatest common divisor of 30 and 42? Suggestion.-Dividing 42 by 30, the remain der is 12; then dividing 30 (the preceding divisor) by 12 (the last remainder) the remainder is 6; finally, dividing 12 (the preceding divisor) by 6 (the last remainder) nothing remains; consequently 6, the last divisor, is the greatest common divisor. Hence, 12)30(2 24 6)12(2 12 96. To find the greatest common divisor of two numbers. Divide the greater number by the less; then divide the preceding divisor by the last remainder, and so on, till nothing remains. The last divisor will be the greatest common divisor. Demonstration.-Since 6 is a measure of the last dividend 12, in the solution above, it must therefore be a measure of the preceding dividend 30; because 30=2× 12+6; now 30 is one of the given numbers. Again, since 6 measures 12 and 30, it must also measure their sum, viz: 30+12, or 42, which is the other given number. (Art. 89. Def. 12. Obs.,) 8. What is the greatest common divisor of 63 and 147 ? 9. What is the greatest common divisor of 91 and 117? 10. Find the greatest common divisor of 247 and 323. 11. Find the greatest common divisor of 285 and 465. 12. What is the greatest common divisor of 2145 and 3471? Of 464320 and 18945? Of 638296 and 33888? Of 18996 and 29932? Of 260424 and 54423? Of 143168 and 2064888 ? 97. To find the greatest common divisor of more than two numbers. First find the greatest common divisor of any two of the given numbers; then, that of the common divisor thus obtained and of another given number, and so on through all the given numbers. The last common divisor found, will be the one required. 13. What is the greatest com. divisor of 63, 105, and 140? 14. Find the greatest common divisor of 16, 24, and 100. 15. Find the greatest common divisor of 492, 744, and 1044. QUEST. Obs. If two numbers have not a common factor, what is true of them as to a common divisor? 96. How find the greatest common divisor of two numbers? 97. Of more than two? |