PROPERTIES OF NUMBERS. FACTORS AND DIVISORS. 119. 1. What two numbers other than itself and 1, multiplied together will produce 6? 8? 10? 12? 2. What two numbers multiplied together will produce 24? 16? 20? 25? 36? 3. What numbers other than itself and 1, will exactly divide 18? 20? 30? 32? 4. What numbers will exactly divide 36? 28? 40? 45? 56? 60? 75? 5. What three numbers multiplied together will produce 24? 27? 36? 50? 6. What are the two smallest numbers other than 1 that will exactly divide 15? 18? 21? 30? 55 ? 7. Which is the largest number other than itself that will exactly divide 27? 42? 35? 48? 8. Name the numbers between 1 and 20 that are the product of two or more numbers greater than 1. 9. Name some numbers that cannot be produced by multiplication except by multiplying the given number and 1 together. 10. Name some numbers that can be produced by multiplying together other numbers than the given number and 1. 11. Of what number are 3 and 4 the factors? 2, 3 and 5? 3, 2 and 7? 2, 4 and 5? 2, 2, 3 and 3? 12. Of what number are 3, 3, 2 and 5 the factors? DEFINITIONS. 120. An Integer, or Integral Number, is a number representing whole units; thus, 5, 7, 12 horses, 16 pounds, are integral numbers. Integers are either even or odd, prime or composite. 121. An Even Number is one that is exactly divisible by 2. All numbers terminating with 0, 2, 4, 6, or 8 are even. 122. An Odd Number is one that is not exactly divisible by 2. All numbers terminating with 1, 3, 5, 7, or 9 are odd. 123. A Prime Number is one that has no integral factors except unity and itself; thus, 2, 3, 5, 11, 17, etc., are prime numbers. 124. A Composite Number is one that has other integral factors besides unity and itself; thus, 15 and 24 are composite numbers, since 155 × 3, and 248 × 3, or 6 × 4, etc. 125. The Factors of a number are the numbers which multiplied together will produce that number (87); thus, 4 and 5 are factors of 20; 3, 4 and 7 of 84. 126. Factoring is the process of separating a number into its factors. 127. An Exact Divisor of a number is one that will divide that number without a remainder; thus, 6 is an exact divisor of 48; and 9 of 72. 1. The exact divisors of a number are also the factors of that number. 2. An exact divisor of a number is sometimes called the measure of that number. 3. When a number is a factor, or divisor of each of two or more numbers, it is called a common factor, divisor, or measure of those numbers. 128. Numbers are prime to each other when they have no common integral factors or divisors; thus, 9 and 16, 12 and 25, are prime to each other. 129. A Prime Factor is one that is a prime number (123). The prime factors of a number are also the prime divisors of it. FACTORING. 130. 1. Name all the even numbers from 0 to 30. 2. Name all the odd numbers from 0 to 30. 5. Name all the composite factors of 30. 6. What are the prime numbers between 20 and 40? 7. What are the prime factors of 21? Of 27? Of 33? 8. Name the composite numbers between 10 and 36. 9. What are the composite factors of 36? 48? 60? 10. What prime factor is common to 20 and 30 ? 11. Name the only prime even number. 12. Name the composite numbers consisting of 1 figure. 13. Name some odd numbers that are prime. 14. Of what number are 3, 3 and 11 prime factors? 15. Of what number are 2, 2, 5 and 7 prime factors? 16. What factors are common to 16 and 24? To their sum? To their difference? 17. What prime factor is common to 15 and 3 times 15? 18. What two composite factors are common to 16 and 3 times 16? 19. What factors are common to the sum and difference of 18 and 30? 20. Name 5 odd numbers of which 3 is a common factor? 21. Name five even numbers of which 5 is a common factor 131. PRINCIPLES.-1. Every prime factor of a number is an exact divisor of that number. 2. A factor of a number is a factor also of any product of that number. 3. A factor common to two or more numbers is a factor also of their sum and of their difference 4. Every composite number is equal to the product of its prime factors. WRITTEN EXERCISES. 132. To resolve a composite number into its prime factors. 1. What are the prime factors of 252 ? 2. What are the prime factors of 126? Of 225? Of 154? RULE.-Divide the given number by any one of its prime factors, and the resulting quotient by another, and so continue to divide until the quotient is a prime number. The several divisors and the last quotient will be the prime factors. PROOF.-The product of all the prime factors will be the 133. 1. What numbers are exact divisors of 21? 36? 2. What number is an exact divisor both of 27 and 63 ? 3. What are the prime divisors of 25? 35? 49? 4. What are the composite divisors of 72? 80? 108? 5. What prime divisor is common to 28, 42 and 56? 6. What is a common measure of 22, 33 and 55 ? 7. What common divisors have 40 and 60? 8. What is the greatest divisor common to 15 and 45? To 26 and 39? To 18, 36 and 72? 9. Name four odd numbers of which 7 is a common divisor. Four even numbers. 10. Name three numbers of which 6 and 9 are common divisors. |