123. Factoring is the process of finding the factors of composite numbers.

18? 20?

28? 30? 45? 49?

124.-1. What are the prime factors of 12? 14? 16? 2. What are the prime factors of 15? 3. What are the prime factors of 21? 4. What are the prime factors of 33? 5. What are the prime factors of 27? 6. What are the prime factors of 36? 7. Name the prime numbers from 1 to 19. to 29.

42? 44?

40? 54?

From 19

8. Name the prime numbers from 31 to 41. From 41 to 83.

9. Name the composite numbers from 4 to 20. From 24 to 36. From 36 to 48. From 48 to 60.

125. Principle.-Every composite number is equal to the product of all its prime factors.

WRITTEN EXERCISES.

126.-1. What are the prime factors of 90?

2. What are the prime factors of 84? 3. What are the prime factors of 75? 4. What are the prime factors of 96?

127. Rule for finding the Prime Factors of a Number.—Divide the given number by any prime number greater than 1 that will divide it without a remainder, and the quotient, if composite, in the same manner; and so proceed until a quotient is obtained which is a prime number.

The last quotient and the several divisors are the prime factors.

129.-1. What integers will divide 15 without a remainder? 21? 35?

2. What integers will divide 16 without a remainder? 27? 42?

3. What integers will divide 44 without a remainder ? 77? 81?

4. What factors have 6 and 24 in common? 15 and 20? 5. What factors have 22 and 25 in common?

6. What prime factors have 12 and 18 in common ? 7. What is the greatest factor common to 12 and 18? 8. What is the product of the prime factors common to 12 and 18?

9. What is the product of the prime factors common to 8 and 24 ?

10. What is the greatest factor common to 8 and 24

DEFINITIONS.

130. A Divisor, or Measure, of a number is any factor of that number.

131. A Common Divisor of two or more numbers is any factor common to those numbers.

132. The Greatest Common Divisor of two or more numbers is the greatest factor common to those numbers.

133. Principle.-The greatest common factor, or divisor, of two or more numbers is equal to the product of all the common prime factors of those numbers.

WRITTEN EXERCISES.

134.-1. What is the greatest common factor or di

visor of 330 and 550?

330 2X3 X5 X 11 550

2 × 5 × 5 × 11

2 X 5 X 11=110

SOLUTION. - Find the prime

factors of the numbers.

The prime factors common to the numbers are 2, 5 and 11, and

their product is 110. Hence, the greatest common factor or divisor is 110.

2. Find the greatest common factor of 27 and 36. 3. Find the greatest common factor of 42 and 35. 4. Find the greatest common divisor of 14, 35 and 63. 135. Rule for finding the Greatest Common Divisor.—Find the prime factors common to the given numbers, and the product of those factors will be the greatest common divisor of the numbers.

PROBLEMS.

136. What is the greatest common factor or divisor of—

1. 26 and 39? 2. 17 and 51 ? 3. 27 and 63?

4. 18 and 96?
5. 45 and 300?
6. 21 and 105?

7. 4, 6 and 18?
8. 12, 30 and 84?

9. 18, 54 and 72?

10. What is the length of the longest pole which will exactly measure 130, 150 or 170 feet?

SECTION XI.

MULTIPLES.

137.-1. Name the numbers from 3 to 15 that will contain 3 without a remainder.

2. What are some of the numbers that are an exact number of times 3?

3. What are some of the numbers that are an exact number of times 5? Of times 8? Of times 9?

4. Of what integers is 12 an exact number of times? 5. Name some number that will contain either 3 or 4 an exact number of times.

6. Name some dividend that will exactly contain and 6. 5 and 7. 6 and 9.

7. What is the least dividend that will contain both 10 and 15 an exact number of times?

8. What are all the prime factors of 10 and 15?

9. What are prime factors of the least number that will exactly contain both 10 and 15?

DEFINITIONS.

138. A Multiple of a number is any number which it will divide without a remainder.

139. A Common Multiple of two or more numbers is any number which each of those numbers will divide without a remainder.

140. The Least Common Multiple of two or more numbers is the least number that each of those numbers will divide without a remainder.

141. Principle. The least common multiple of two or more numbers is the least number that contains all the prime factors of those numbers.

WRITTEN EXERCISES.

142.-1. What is the least common multiple of 4, 12

and 30?

4=2×2.

12 2X2 × 3. 30=2X3 X 5.

2 X2 X3 X5

60.

SOLUTION.-Find the prime factors of the given numbers.

The multiple of 4 must contain the prime factors 2 and 2; the multiple of 12, the additional prime

factor 3; and the multiple of 30, the additional prime factor 5. These factors are 2, 2, 3 and 5, and their product is 60. Therefore, the least common multiple of 4, 12 and 30 is 60.

2. Find the least common multiple of 8 and 12. 3. Find the least common multiple of 4, 6 and 20. 143. Rule for finding the Least Common Multiple.-First find the prime factors of the given numbers. The product of these different prime factors, each factor being taken the greatest number of times it occurs in any of the numbers, will be their least common multiple.