152. Prime Factors are factors which are prime num bers. Thus, 2, 2, and 3 are the prime factors of 12. WRITTEN EXERCISES. 153. 1. Write all the prime numbers between 1 and 25; 25 and 50; 50 and 75; 75 and 100. 2. What are the prime factors of 56? EXPLANATION.-We divide the number by any prime factor; then divide the quotient by any prime factor, etc., until the quotient 1 is obtained. The several divisors are the prime factors required. Hence, the answer is 2, 2, 2 and 7. What are the prime factors of: OPERATION. 2)56 2)28 2)14 7) 7 154. A Multiple of a number is a number which ex actly contains it. Thus, 24 contains 6 exactly 4 times, hence 24 is a multiple of 6. Is 16 a multiple of 8? 25 a multiple of 5? 27 of 7? 36 of 9? 42 of 6? 54 of 9? 63 of 7? 64 of 10? 72 of 12? 155. A Common Multiple of two or more numbers is a number which exactly contains each of them. Thus, 12 is a common multiple of 2, 3, 4, and 6; 18 of 2, 3, 6, and 9. N. I.-9. EXERCISES. 1. Name three common multiples of 2, 3, and 6. Ans. 6, 12, and 18. 2. Name three common multiples of 4, 5, and 10. Ans. 20, 40, 60. 3. Name three common multiples of 3, 4, 12. Ans. 12, 24, 36. 4. Name three common multiples of 7 and 2; 5 and 8; 3, 7, and 2; 5 and 12; 3, 5, and 10; 2, 4, and 8. 156. The Least Common Multiple of two or more numbers, denoted by L. C. M., is the least number that will exactly contain each of them. Answer these questions by referring to the preceding exercises: What is the L. C. M. of 2, 3, and 6? Of 4, 5, and 10? Of 3, 4, and 12? What is the L. C. M. of 5 and 6? 6 and 7? 6 and 8? 5 and 9? 7 and 10? 2, 3, and 5? 157. PRINCIPLE.-The c factors of the L. C. M. of two or more numbers are all the prime factors of each. Thus, the prime factors of 12 are 2, 2, and 3. Taking out the factor 2 from each, then the factor 3 from the first and second, then the factor 5 from the second and third, there are left, 2 in the first, and 5 in the third. Hence, the c factors of the L. C. M. of 12, 30, and 50 are 2, 3, 5, 2, and 5. WRITTEN EXERCISES. 158. What is the L. C. M. of 12, 15, and 25? EXPLANATION.-Divide any two or more of the numbers by any common prime factor, as 3, and bring down with the quo OPERATION. 3)12 15 25 5) 4 5 25 4 1 5 tients 4 and 5 such numbers (25) as do not contain the divisor. Again, divide out by 5, as it is a prime factor common to 5 and 25, and bring down the 4. Now there is no factor except 1 that will divide two of the numbers 4, 1, and 5. Hence, the two divisors 3 and 5, and the quotients 4 and 5 are the c factors of the L. C. M.; that is, the L. C. M. is 3 X5 X 4 X 5 contains all the prime factors of 12, 15, and 25. Find the L. C. M. of: = 300, as it Ans. 12; 45; 100. 2. 6 and 12; 9 and 15; 20 and 25. 9. 5, 7, 11, and 15; 3, 7, 13, and 39. 10. 4, 5, 6, 10, 12, 15, 20, and 30. Ans. 30; 60. Ans. 120; 160. Ans. 1800; 480. Ans. 1155; 273. 159. QUESTIONS FOR REVIEW. Ans. 60. What is: 1. A divisor of a number? 2. A common divisor of two or more numbers? 3. The G. C. D. of two or more numbers? Give an example of each. What is 1. A prime number? 2. A composite number? 3. A prime factor? 4. A multiple of a number? 5. A common multiple of two or more numbers? 6. The L. C. M. of two or more numbers? Give an example of each. What is the principle of: 1. The G. C. D.? 2. The L. C. M.? When is a number exactly divisible by: 2? 3? 4? 5? 6? 8? 9? 10? COMMON FRACTIONS. INDUCTIVE EXERCISES. 160. When an apple, an orange, a number, or a bar of soap is divided into two equal parts, what is each part called? How is 1-half Ans. Divide it into two equal parts, and take one of the parts. What is of a pile of 2 books? of $2? of $4? of $16? When an apple, an orange, a number, or a bar of soap is divided into three equal parts, what is each part called? What are two of the parts called? How are 1-third and 2-thirds written? Ans. and . How do we get of a thing? Ans. Divide it into three equal parts and take one of the parts. What is of a pile of 3 books? of $3? of $6? of $12? of 24c.? of 60 bushels? How do you get of a thing? Ans. Divide it into three equal parts and take one of the parts 2 times. What is of a pile of 3 books? of $3? % of $12? of 27c.? When an apple, an orange, a number or a bar of soap is divided into four equal parts, what is each part called? What are two of the parts called? What are three of the parts called? How are 1-fourth, 2-fourths, and 3-fourths written? Ans. 1, 4, and 2. How do we get of a thing? Ans. Divide it into four equal parts, and take one of the parts. What is of a pile of 4 books? of $4? of $20? of 40c.? How do we get of a thing? Ans. Divide it into four equal parts and take 1 part 2 times. How much is of a pile of 4 books? 4 of 4? of 16? 4 of 32? How do we get of a thing? Ans. Divide it into four equal parts, and take 1 part 3 times. How much is of a pile of 4 books? of 4? of 12? of 40? DEFINITIONS. 161. Fractional Parts are parts obtained by dividing any thing, or a unit, into any number of equal parts. Thus: halves, thirds, fourths, fifths, sevenths, tenths, etc., are fractional parts. 162. A Fractional Unit is one of the equal fractional parts into which a thing is divided. |