11. 5, 2, and what other number are the prime factors of 70? 12. Give two factors of 30 that are not prime. 13. What even number is prime? 84. Rule for finding whether a Number is Prime or Composite. 1. If the given number is odd, divide it by 3. 2. If 3 gives a remainder, divide the given number by 5. 3. Continue this process, using each prime number in order as a divisor, until an exact divisor is found, or until the divisor equals or exceeds the quotient. If no exact divisor is found until the divisor used equals or exceeds the quotient, the number is prime. Otherwise it is composite. Even numbers need not be tested. Why? 85. 1. Applying the tests given on page 26 instead of actually dividing by 3 and 5, determine whether 191 is prime or composite. Since 191 is not divisible by 3 or 5. (How do we know?) the divisor, 17, is greater than the quotient, 11, and no exact divisor has been found, 191 must be prime. Find whether each of these numbers is prime or composite: 86. Written 1. Find the prime factors of 7020. 21 7020 2 3510 By what kind of numbers do we divide? Why? What besides the divisors is a prime factor? 87. Dividing both dividend and divisor by the same number affects the quotient how? In either method we divide both dividend and divisor by 2, by 3, and by 11. Taking out the same factor from both dividend and divisor is cancellation. 88. Solve by cancellation: 1. Divide 36 × 54 × 49 × 38 × 50 by 70 × 18 × 30. 2. (28 × 152 × 48) ÷ (14 × 19 × 24 × 2 × 8) = ? 3. (182 × 5 × 54) ÷ (13 × 35 × 6) = ? 4. What is the quotient of 108 × 48 × 80 divided by 27 × 72 × 40 ? 5. Divide 125 × 45 × 7 × 10 by 49 × 5 × 2 × 225. 6. Divide 65 x 51 × 11 × 9 × 4 by 17 × 20 × 12 × 11 × 26. 7. Divide 25 × 26 × 72 × 14 by 78 × 9 × 120. 8. How many bushels of potatoes at 80 cents a bushel must be given in exchange for 45 pounds of tea at 64 cents a pound? 9. (240 × 36 × 385 × 26) ÷ (12 × 154 × 65). 10. What prime factor besides 19 and 11 has 8569? 3. Name all the numbers of which 24 is a multiple. 4. Define multiple. 90. A number that exactly contains two or more numbers is a common multiple of those numbers; e.g. 12 is a common multiple of 2, 3, 4, and 6. 36 also is a common multiple of 2, 3, 4, and 6. Can you name any other common multiple of 2, 3, 4, and 6? 91. The smallest number that exactly contains two or more numbers is their least common multiple (L. C. M.); e.g. 18 is the least common multiple of 3, 6, and 9. 36 is a common multiple of 3, 6, and 9. Why is it not the least common multiple? 93. When the least common multiple is a large number, the following direct method is employed in finding it : Let it be required to find the L. C. M. of 12, 15, and 18. What kind of factors have we found? A number, in order to contain 12, must have what prime factors? What prime factors must it have in order to contain 15? 18? A number that contains 12, 15, and 18 must have how many factors 2? How many factors 3? How many factors 5? What is the smallest number that has the factors 2, 2, 3, 3, and 5? What, then, is the L. C. M. of 12, 15, and 18? The prime factors may be easily found in this way: 95. A number that will exactly divide two or more numbers is a common divisor of those numbers; e.g. 5 is a common divisor of 30, 40, and 60. 96. The largest number that will exactly divide two or more numbers is their greatest common divisor (G. C. D.); e.g. 10 is the greatest common divisor of 30, 40, and 60. NOTE. A common divisor is sometimes called a common factor, and the greatest common divisor is sometimes called the highest common factor. 97. Numbers that have no common divisor are prime to each other; e.g. 13 and 15. |