COR.-When a number is resolved into its prime factors, the original number is divisible by all these prime factors, and by all the quotients arising from these factors as divisors of the original number; and when a factor occurs more than once, by the products of the like factors. DEF.-One number is called a Multiple of another number, when it is exactly divisible by that other number. Find all the numbers of which the following numbers are multiples : FACTORS. DIVISORS. 45. 3, 3, 5. 3, 5, 9, 15. 60. 2, 2, 3, 5. 2, 3, 5, 4, 6, 10, 12, 15, 20, 30. 70. 2, 5, 7. 2, 5, 7, 10, 14, 35. 12. 2, 2, 2, 3, 3. 2, 3, 6, 8, 9, 12, 18, 36. 175. 3, 5, 5. 3, 5, 15, 25. 80. 2, 2, 2, 2, 5. 2,5, 4, 8, 10, 16, 20, 40. 100. 2, 2, 5, 5. 2, 4, 5, 10, 20, 25, 50. PROBLEMS. 3, 5. 1. Find the least common multiple of 9 and 15. 9 15 It is evident that any number which 3, 3. contains all the prime factors of each 3 X3 X5 = 45. 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 con tains all the prime factors of 6; hence, when any given number is a multiple of another given number, the later may be canceled. 3. Find the least common multiple of 6, 8, 10, 12, 15. 2 8 10 12 15 As 12 is a multiple of 2 4 5 6 15 6, cancel the 6; and as 2 is a common factor of 3 15 2 x 2 x 2 x 15 = 120 two or more numbers, divide the 2 into them, reserving the divisor, the quotients, and the numbers not divided; as 15 is a multiple of 5, cancel 5; and as 2 is now a common factor of two of the quotients, divide it into them, reserving as before ; and as 15 is now a multiple of 3, cancel 3; the product of all the divisors and of the remaining quotients and original numbers, if there be any, will be the least common multiple. EXAMPLES. 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. %. 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. PROBLEMS. Find the greatest common divisor of the following numbers: 1. Of 6 and 14. 2, 3. 2, %. The only factor common is 2 = G. C. D. 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. The process of the last example is the shortest foi 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 Divide the smaller number 84 into the larger, and the re63 ) 84 (1 mainder into the last divisor; 63 and again the remainder into 21 ) 63 ( 3 the last divisor, until there is no remainder; the last divi63 sor 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. 6. Find the G. C. D. of 323 and 425. G. C. D = 17. 1. Find the G. C. D. of 2310 and 4626. G. C. D. = 6. 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 |