INDUCTIVE EXERCISES. 88. 1. What numbers are exact divisors of 21? 36? 6. What factors or divisors are common to 12, 24, and 36? 7. What two numbers will exactly divide or measure 24 miles? 35 acres? 48 pounds? 8. What two numbers will exactly measure 10 and 20? Their sum? Their difference? Their product? 9. What is the greatest exact divisor of the sum and difference of 16 and 24? Of 18 and 45? 10. What is the G.C.D. of 28, 35, and 42? 11. Name three numbers of which 9 is the G.C.D. 12. What is the G. C. D. of 22, 44, and 66? 13. What divisor is common to 6 and 9? To 6 and 9, and to 54, their product? To 8 and 12, and to 96, their product? 14. What divisor is common to 18 and 27, and to 45, their sum? To 18 and 27, and to 9, their difference? PRINCIPLES.-I. Every prime factor of a number is an exact divisor of that number. II. The G. C.D. of two or more numbers is the product of all their common prime factors. 89. A common divisor of two or more numbers is a common factor of each of them. 90. The greatest common divisor of two or more numbers is the greatest common factor, and is the product of all the common prime factors. 91. To find the G.C.D. of two or more numbers. 1. What is the G.C.D. of 18, 30, and 42? EXPLANATION. By factoring, we find 2 and 3 to be the prime factors common to all the given numbers. The product of these prime divisors, 2 × 3 = 6, is the G.C.D. of 18, 30, and 42 (PRIN. II). In like manner, find the G. C. D. 1ST METHOD. 2|18 30 42 3 9 15 21 3 5 . EXPLANATION.-Draw two vertical lines, and place th greater number on the right, and the less on the left, one line lower down. Divide 323 by 247, and write the quotient 1 between the vertical lines, the product 247 under the greater number, and the remainder 76 below. Next, divide 247 by this remainder 76, writing the quotient 3 between the vertical lines, the product 228 on the left, and the remainder 19 below. Again, dividing the last divisor, 76, by 19. there is no remainder. Hence, 19, the last divisor, is the G.C.D. of 247 and 323. In like manner, find the G. C.D. of Resolve the given numbers into their prime factors; the product of all the common prime factors is the G. C.D. 2d. By continued division: 1. Divide the greater number by the less, and the less number by the remainder, if any, and so continue to divide the last divisor by the last remainder, till nothing remains. The last divisor will be the G. C.D. 2. If more than two numbers are given, find the G. C.D. of two of them, then of this divisor and a third number, and so on. If the last divisor is 1, the numbers are prime to each other, and have no common divisor greater than 1. Find the greatest common divisor 15. Of 432 and 648. 16. Of 135 and 315. 17. Of 182 and 364. 18. Of 756 and 1575. 19. Of 1008 and 1036. 20. Of 216, 360, and 432. 21. Of 84, 126, and 462. 22. Of 141, 799, and 940. 23. Of 3281 and 10778. 25. What is the greatest number that will divide 620, 1116, and 1488? 396, 5184, and 6914? 26. A speculator has 3 fields, the first containing 18, the second 24, and the third 40 acres, which he wishes to divide into the largest possible lots having the same number of acres in each. How many acres in each lot? 27. The Erie R. R. has 3 switches, or side tracks, of the following lengths: 3013, 2231, and 2047 feet. What is the length of the longest rail that will exactly lay the track on each switch? 28. A farmer wishes to put 336 bushels of corn and 812 bushels of wheat into the least number of bins possible, that shall contain the same number of bushels, without mixing the two kinds of grain. How many bushels must each bin hold? 29. Three persons have respectively $630, $1134, and $1386, with which they agree to purchase horses, at the highest price per head that will allow each man to invest all his money. How many horses can each man buy? 92. 1. What numbers between 5 and 30 are exactly divisible by 4? By 5? By 6? By 8? 2. Name some numbers of which 5 is a factor. Of which 6 is a factor. 3. What prime factors are common to 6 and 5 times 6? 4. Name some numbers that are exactly divisible by 3 and 5. 4 and 6. 3 and 7. 5. A number which contains another number one or more times is called a inultiple of that number. 6. By what three prime numbers can 42 be divided? 7. What numbers less than 40 are exactly divisible by 7? 8. What numbers from 4 to 36 have both 2 and 3 as factors? Have 2 and 5 as factors? 9. What is the least number of which 3, 4, and 5 are factors? 10. Name two numbers of which 6 and 9 are factors. Name the least number of which they are common factors. 11. Name three multiples of 3? 4? 5? 6? 8? PRINCIPLES.-I. Every multiple of a number contains all the prime factors of that number. II. The L. C. M. of two or more numbers contains all the prime factors of each of those numbers. III. If two or more numbers are prime to each other, their product is their L. C. M. 93. A multiple of a number is any number of times that number. 1. A multiple is necessarily composite; a divisor may be either prime or composite. 2. A number is a divisor of all its multiples, and a multiple of all its divisors. 94. A common multiple of two or more numbers is a number exactly divisible by each of them. 95. The least common multiple (L.C.M.) of two or more numbers is the least number exactly divisible by each of them. Two or more numbers can have but one least common multiple. 96. To find the L.C.M. of two or more numbers. 1. Find the L. C. M. of 14, 21, and 45. EXPLANATION.-Resolving the given numbers into their prime factors, we find the L. C. M. must contain the factor 2 once, 7 once, 3 twice, and 5 once, and no other factors (PRIN. II). Hence, 2 × 7 × 3 × 3 × 5630, the L. C. M. EXPLANATION.-Divide the given numbers 21 and 45 by the prime factor 3, and write the quotients and undivided number 14 in the second line. In the same manner, divide successively by the prime factors 7 and 3. The last quotient, 5, being a prime number, is not divisible. Hence, 3 × 7×3×2×5=630, L. C. M. (PRIN. III). Find the L.C.M. 4. Of 16, 48, and 108. FIRST METHOD. 14=2×7 7x3 3x3x5 2x7×3×3×5 21= 5. Of 14, 42, and 63. RULE.-Resolve each of the numbers into their prime factors; the product of the highest power of each prime factor is the L. C.M. Or, |