85.

2.

EXERCISES.

1. Learn the prime numbers from 1 to 100.

Write the prime factors of the other numbers from 51 to 100.

3. Remember the squares of all the numbers from 13 to 25, inclusive, 132-169, etc.

4. Remember all cubes as far as 93, at least.

86. 1. Of what numbers is 15 a multiple? 18?

20? 24?

30?

2. Any number is a multiple of any one of its factors.

3. Name two multiples of both 3 and 4; 3 and 5; 6 and 9; 3, 4, and 6; 2, 6, and 9; 5, 6, and 10; 4, 7, and 14. Are these all the multiples they can have?

4. A common multiple of several numbers is a number which will contain each of them.

5. What is the least number that is a multiple of 3 and 4? Of 3 and 5? Of 6 and 9? Of 3, 5, and 10? Of 6, 9, and 18? What is such a multiple called?

6. The least common multiple of several numbers is the least number that will contain each of them.

7. Name two multiples of 6 and 10. Have 6 and 10 any prime factors not found in these multiples? Name two multiples of 4, 6, and 9. Have they any

prime factors not found in the multiples?

8. A common multiple of several numbers contains all the prime factors found in each of them.

9. What is the least common multiple of 6 and 10? Has it any prime factors not found in 6 and 10? What is the least common multiple of 6, 10, and 9? Has it any prime factors not found in 6, 10, and 9?

10. The least common multiple of several numbers contains no prime factors except those found in the numbers.

11. Find the least common multiple (L. C. M.) of 50, 40, 36, and 25.

50=2 X52

40=23×5

36=22×32

25

L. C. M. 2×52X22X32=1800

EXPLANATION: Resolve the numbers 50, 40, and 36 into their prime factors. Do not resolve 25 into its prime factors; as it is a factor of 50, it will be a factor of any multiple of 50, by

Principle I. Write L. C. M., placing after it the sign of

equality. The L. C. M. must be a multiple of 50; hence, it must contain the prime factors of 50, which are 2 and two 5's. Write these factors after the sign of equality. The L. C. M. must be a multiple of 40 also. It must, therefore, contain the prime factors of 40, which are three 2's and 5. The 5 and one 2 have already been written as factors of the L. C. M., so write two 2's more as factors of the L. C. M. The L. C. M. must be a multiple of 36 also. Therefore, it must contain the prime factors of 36, which are two 2's and two 3's. Two 2's are already among the factors of the L. C. M.; hence, write only the two 3's as its factors. There is now indicated a number which will contain 50, 40, or 36. Because it contains the prime factors of these numbers, it is their common multiple; and because it contains no prime factors but those found in the numbers, it is their L. C. M.

The product of these factors is 1,800, the required L. C. M. (In finding this product, observe that 22×522=100.)

87. Rule for Finding the Least Common Multiple of Several Numbers.

Factor the several numbers; then, to make their least common multiple, take all the prime factors found in each of them; multiply together these prime factors only.

EXERCISES.

88. Find the L. C. M. of the following:

25. What is the least number of apples that can be equally distributed among 8, 12, 16, or 20 boys?

26. What is the least sum of money that can be expended for sheep, cows, or horses at $9, $39, and $65 each, respectively?

27. What is the shortest piece of cord that can be cut exactly into pieces either 10, 12, 15, 16, or 18 feet long?

28. What are the fewest words that will make an integral number of lines across the page, whether the lines average 8, 9, or 10 words?

would there be in each case?

89.

How many lines

GREATEST COMMON DIVISOR.

1. Name all the divisors (factors) of 6, 8, 9, 10, 12, 14, 15, 16, 20, 24, 25.

2. Name a number that is a divisor of 15 and 35; 28 and 49; 36 and 63; 33 and 88; 9, 15, and 21; 14, 21, and 35; 15, 20, and 25. What is such a number called?

3. A common divisor of several numbers is a number that is a divisor of each of them.

4. Name all the common divisors of 12 and 18; 10 and 20; 15 and 30; 24 and 36; 6, 12, and 18; 16, 24, and 32; 18, 27, and 36; 24, 36, and 48. Which is the greatest divisor in each case? What is such a number called?

5. The greatest common divisor of several numbers is the greatest number that is a divisor of each of them.

6. Name all the prime factors of 42. What is the product of any two of them? Is this product a divisor of 42?

7. Name all the prime factors of 60. What is the product of any three of them? Is this product a divisor of 60?

8. Any number is a multiple of the product of any of its prime factors.

9. Name all the common prime factors of 24 and 36. What is the product of any two of them? Is this product a divisor of 24 and 36?

10. Name all the common prime factors of 30 and 60. What is the product of any two of them? Is this product a divisor of 30 and 60?

11. A common divisor of any two numbers is one of their common prime factors or a product of two or more of them.

12.

What is the product of all the common prime factors of 24 and 36? Is this product a divisor of 24 and 36? Have they a larger common divisor?

13. What is the product of all the common prime factors of 30 and 60? Is this product a divisor of 30 and 60? Have they a larger common divisor?