45. 60. 70. 72. 75. 80. 100. Find all the numbers of which the following numbers are multiples: FACTORS. 3, 3, 5. 2, 2, 3, 5. 2, 5, 7. 2, 2, 2, 3, 3. 3, 5, 5. 2, 2, 2, 2, 5. 2, 2, 5, 5. 9 15 3, 3. 3, 5. 3x3x5 = 45. DIVISORS. 3, 5, 9, 15. 2, 3, 5, 4, 6, 10, 12, 15, 20, 30. 2, 5, 7, 10, 14, 35. 2, 3, 6, 8, 9, 12, 18, 36. 3, 5, 15, 25. 2, 5, 4, 8, 10, 16, 20, 40. PROBLEMS. 1. Find the least common multiple of 9 and 15. It is evident that any number which contains all the prime factors of each number is a common multiple of the given numbers; and the least common multiple must contain these factors and no others; and if the same factor occur several times in any number, it must occur just as often in the multiple. 2. Find the least common multiple of 6 and 12. 6 12 As twelve is a multiple of 6, it contains all the prime factors of 6; hence, when any given number is a multiple of another given number, the later may be canceled. REM.-Since any factor that is common to two or more given numbers is only taken once, the solution may be shortened, as in Ex. 3. Since 2 is common to several given numbers, the factor may be taken from all and placed as a divisor; 2 is again common, so also 3 and 5; these being set aside, there remain besides only 2; consequently, 2 × 2 × 3 × 5 × 2 = 120, is the least common multiple. 1. Find the least common multiple of 6 and 15. Ans. 30. 2. Find the least common multiple of 6, 15, and 42. Ans. 210. 3. Find the least common multiple of 10, 12 and 14. Ans. 420. 4. Find the least common multiple of 4, 6, and 8. Ans. 24. 5. Find the least common multiple of 6, 8, and 10. Ans. 120. 6. Find the least common multiple of 12, 18, 27, and 36. Ans. 108. 7. Find the least common multiple of 10, 15, 25, and 40. Ans. 600. 8. Find the least common multiple of 14, 18, 21, and 28. Ans. 252. 9. Find the least common multiple of 15, 25, 36, and 48. Ans. 3600. 10. Find the least common multiple of 25, 45, 70, and 90. Ans. 3150. 11. Find the least common multiple of 4, 8, 16, 32, and 64. Ans. 64. 12. Find the least common multiple of 3, 7, 11, and 13. Ans. 3003. GREATEST COMMON DIVISOR. The Greatest Common Divisor of two or more numbers, is the highest number that will exactly divide the numbers. COR.-The greatest common divisor of two or more numbers must be the factor or the product of all the factors which are common to the given numbers. Find the greatest common divisor of the following numbers: 1. Of 6 and 14. 2, 3. 2, 7. The only factor common is 2 = G. C. D. 3. Of 42 and 70. PROBLEMS. 2. Of 12 and 18. 12 = 2, 2, 3, and 18 = 2, 3, 3. One 2 and one 3 are common: therefore, 2 x 3 = 6 = G. C. D. 84 126 42 63 14 2 42 2, 3, 7, and 70 2, 5, 7; = = 2×7 = 14 G. C. D. 210 105 21 35 3 4. Of 84, 126, and 210. By this method the common factors are readily found. 2×3×742. The process of the last example is the shortest for finding the greatest common divisor of small numbers; but when the numbers are large and the common factors not so readily found, the following method is generally adopted. 5. Find the G. C. D. of 84 and 147. 84) 147 ( 1 84 63) 84 (1 63 21) 63 ( 3 Divide the smaller number into the larger, and the remainder into the last divisor; and again the remainder into the last divisor, until there is no remainder; the last divisor is the G. C. D. of the two numbers. ANALYSIS. As each number is a multiple of the G. C. D., so the difference of the numbers is also a multiple of the G. C. D.; hence in every case the dividend and divisor are multiples of the G. C. D., and whenever the divisor is contained in the dividend, that divisor is the G. C. D. G. C. D 17. 6. Find the G. C. D. of 323 and 425. CANCELLATION. THEOREM. The dividend contains all and exactly the same factors as the divisor and quotient. Any composite number is the product of all its prime factors, and may be resolved into them. The prodnct of any two integral numbers is a composite number and must contain all the factors of both numbers; and as a dividend is the product of its divisor and quotient, it must contain the same factors as its divisor and quotient. COR. 1.-The same is true if one or both divisor and quotient be fractional; for when reduced to a common denominator, their numerators may be regarded as integral. COR. 2.-Every factor of the divisor will cancel the same factor in the dividend. COR. 3. The factors which are not canceled by those of the divisor will be the factors of the quotient. COR. 4.-Canceling a factor in the dividend divides. the quotient by the same factor. COR. 5.-Canceling a factor in the divisor multiplies the quotient by the same factor. 3. Divide 500 by 100. 500 = 18, Ans. = 5, Ans. = 5, Ans. 4. A man bought 30 yards of cloth at $5 a yard; he then exchanged it for other cloth at $3 a yard. How many yards of the latter did he get? |