11. What are the prime factors of 441 ? 12. What are the prime factors of 315? 13. What are the prime factors of 1320 ? 14. What are the prime factors of 1728? 15. What are the prime factors of 1600? 16. What are the prime factors of 8424? 17. What are the prime factors of 4284? 18. What are the prime factors of 8364? 19. What are the prime factors of 7698? 20. What are the prime factors of 1682? 21. What are the prime factors of 2585? 22. Prove that 907 is prime.

OPERATION.

31) 907 (29

62

287

From Art. 94, a, b, and d, we see that 2, 3, and 5 are not factors of 907. We then try every other prime number as a divisor to 31, inclusive, and find 907 is not divisible by any of them. There can be no prime factor in 907 greater than 31; for if there were, the quotient would be another factor less than 29; but we have found above that there was no factor in 907 less than 29. Therefore 907 is prime.

279

8

23. Prove that 743 is prime.

24. Prove that 997 is prime.

25. Prove that 1997 is prime.

GREATEST COMMON DIVISOR.

96. Oral Exercises.

26. Find a factor common to 8 and 12.
27. Find a factor common to 6, 9, and 15.
28. Find a factor common to 24 and 36.
29. Find a factor common to 18, 27, and 45.

30. What is the greatest factor common to 4, 8, and 12?

31. What is the greatest factor common to 8, 16, and 24? NOTE. A common factor is generally called a common divisor, and the greatest common factor the greatest common divisor.

97. A Common Divisor of two or more numbers is a number that will divide each of them without remainder; thus, 3 is a common divisor of 12, 18, and 30.

98. The Greatest Common Divisor of two or more numbers is the greatest number that will divide each of them without remainder; thus, 6 is the greatest common divisor of 12, 18, and 30.

99. Written Exercises.

32. What is the greatest common divisor of 18, 30, and 48?

=

OPERATION.

18 2×3×3

30 2 X3 X5

48 2 X3 X 2 × 2 × 2

Ans. 2 X 3

= 6.

We see that 2 and 3 are factors common to all the numbers, and, furthermore, they are the only common factors; hence their product, 2 × 3 =6, is the greatest common divisor of the given numbers.

33. What is the greatest common divisor of 60, 72, 48, and 84?

to find the greatest common divisor of two or more numbers,

Rule 1.

Separate each number into its prime factors, and the continued product of all the prime factors that are common to all the given numbers, each factor taken the least number of times that it occurs in any of the numbers, will be the common divisor sought.

34. What is the greatest common divisor of 24, 40, 64, 80, 96, 120, and 192? Ans. 2 x 2 x 2 = 8.

NOTE 1. The greatest common divisor cannot be greater than 24; and as 24 is a divisor of 96, 120, and 192, the answer is not affected by the addition of 96, 120, and 192; also, 40 is a divisor of 80. The question then is at once reduced to this, - What is the greatest common divisor of 24, 40, and 64?

35. Find the greatest common divisor of 15, 45, 75, 105, 135, 150, and 300. Ans. 15. 36. Find the greatest common divisor of 25, 45, and 70. 37. Find the greatest common divisor of 24, 36, and 64. 38. Find the greatest common divisor of 24, 48, 72, and 88.

NOTE 2. As in Ex. 3, this question becomes, What is the greatest common divisor of 24 and 88?

39. Find the greatest common divisor of 45, 75, 90, 135, 150, and 180.

45 and 75? Apply this principle whenever you can.

40. I have three house lots, the first 136 feet front, the second 184 feet front, and the third 216 feet front; how long is the longest rail which will exactly fence the front of each lot? How many lengths of fence will be required for each lot? Ans. 8 feet.

100. When the given numbers are not readily separated into their prime factors, their greatest common divisor can be found by

Rule 2.

Of two numbers divide the greater by the less, and, if there is a remainder, divide the divisor by the remainder, and continue dividing the last divisor by the last remainder until nothing remains; the last divisor is the greatest common divisor of the two numbers.

If more than two numbers are given, find the greatest divisor of two of them, then of this divisor and a third number,

and so on until all the numbers have been taken; the last divisor will be the divisor sought.

41. What is the greatest common divisor of 18 and 32?

OPERATION.

18) 32 (1

18

14) 18 (1
14

4) 14 (3

12

Ans. 2) 4 (2

4

0

(For explanation of this method, see Appendix, page 338.) 42. What is the greatest common divisor of 426 and 784?

Ans. 2.

43. What is the greatest common divisor of 1215 and 1878? 44. What is the greatest common divisor of 1071 and 1870? 45. What is the greatest common divisor of 3696 and 1440? 46. What is the greatest common divisor of 6237 and 2520? 47. What is the greatest common divisor of 16, 24, and 36?

48. What is the greatest common divisor of 42, 48, 216, and 234?

49. What is the greatest common divisor of 154, 210, and 420?

50. What is the greatest common divisor of 18 and 11?

51. What is the greatest common divisor of 36, 60, and 140 ?

LEAST COMMON MULTIPLE.

101. Oral Exercises.

52. What whole numbers less than 45 will contain 8?

53. What whole numbers less than 50 will contain both 6 and 9?

54. What whole number less than 75 will contain 4, 6, and 10? 55. What is the least number that is divisible by 6 and 8? 56. What is the least number that is divisible by 2, 3, 4, and 6?

57. What is the least number that is divisible by 4, 6, and 8?

102. A Multiple of a number is any number which is divisible by that number; thus, 15 is a multiple of 5 and also of 3; 21 is a multiple of 7 and of 3.

NOTE. Every number is both a divisor and a multiple of itself.

103. A Common Multiple of two or more numbers is any number which is divisible by each of the given numbers; thus, 48 is the common multiple of 4, 6, and 8.

104. The Least Common Multiple of two or more numbers is the least number that is divisible by each of the given numbers; thus, 24 is the least common multiple of 4, 6, and 8.