(7 x 8)-(5 × 5) means that the product of 7 and 8 is to be diminished by the product of 5 and 5. Thus, 5 · 3 × 3 or [5–3]×3 may be used instead of (5 – 3) × 3. 85. In finding the value of expressions in which the parenthesis, vinculum, or brackets are used, First, perform the operations on the numbers that are written within parentheses and brackets, or under the vinculum, as indicated by the signs. Next, multiply and divide, as indicated by the signs and ÷. Finally, add and subtract, as indicated by the signs + and 5. 4164. 6. (416) 4. (3 × 4)-(2 × 3). 3 12. [(5+6) × 4 − 5 × 8] × 9. 13. (3 × 4)× 5−(9+7) ÷ 8. 7. (3+4) x (8-5). 14. 2+12÷4+(8+8÷4)÷2. 15. (312+36)-(381-215) — 65. 16. (214-81)-(115-18+6) + 10. 17. (413-200) (118-244 + 6) + 3. 18. (171-86) - (3 x 4 +27) +10. 22. (99-3)÷ 8 − (86 + 10) ÷ 12 + (3 + 6) ÷ 3. FACTORING. 86. 1. What is the product when 3 and 2 are multiplied together? What are 3 and 2 of their product? (Art. 86.) 2. What factors will produce 9? 3. What numbers when multiplied together will produce 12? What are 3 and 4, or 6 and 2 of their product? 4. What are the factors of 20? 5. What are the factors of 27 ? 6. What are the factors of 21 ? 7. What are the factors of 63? called of 63 besides a factor? Of 36? Of 15? Of 25? Of 32 ? Of 50? What else may 7 be 8. If 9 is one of two factors of 18, what is the other factor? If 3 is one of the factors? If 6 is one of the factors? 9. What numbers will exactly divide 18? 25? 36? 10. Give the exact divisors of 42; 96; 35; 50; 27; 72. 11. Give the factors of 36; 40; 48; 70; 80. 12. Give the exact divisors of 44; 56; 64; 84; 96. 13. Name the exact divisors of 49; 88; 63; 24; 27. 14. What numbers between 0 and 10 cannot be exactly divided by any number except themselves and 1? What numbers between 10 and 20? Between 20 and 30? 15. What numbers between 0 and 10 can be exactly divided by other numbers besides themselves and 1? Between 10 and 20? Between 20 and 30? 16. Select from the following the numbers that have no exact divisors except themselves and 1: 35, 42, 63, 56, 61, 47, 49, 81, 37, 26, 18, 45. 70 17. Select from the following the numbers that have exact divisors besides themselves and 1: 24, 36, 41, 39, 27, 45, 33, 37, 50, 44, 60, 71, 72. 87. A number that expresses whole units is called an Integer. Thus, 5, 27, 35 are integers, or integral numbers. 88. The integers which, upon being multiplied together will produce the number, are called Factors of the number. Thus, 5 and 3 are the factors of 15. 89. An integer which will divide a number without a remainder is called an Exact Divisor of the number. Thus, 2, 3, 6, and 9 are exact divisors of 18. They are also factors of 18. 90. A number that has no exact divisors except itself and 1 is called a Prime Number. Thus, 1, 3, 5, 7 are prime numbers. 91. A number that has exact divisors besides itself and 1 is called a Composite Number. Thus, 24, 36, 40, 100 are composite numbers. 92. Factors that are prime numbers are Prime Factors. Thus, 7 and 5 are the prime factors of 35. 93. A number that is exactly divisible by 2 is called an Even Number. Thus, 8, 12, 20, 24 are even numbers. 94. A number that is not exactly divisible by 2 is called an Odd Number. Thus, 15, 21, 35, 43 are odd numbers. 95. The process of separating a number into its factors is called Factoring. TESTS OF DIVISIBILITY. 96. 1. Make a list of numbers from 1 to 126, which have 2 for one or more of their factors or divisors. Notice what the right-hand, or units' figure, of each is. 2. Make a list of numbers from 1 to 100 which have 5 for one or more of their factors or divisors. Notice what the units' figure of each is. 3. Make a list of numbers from 1 to 100 which have 3 for one or more of their factors or divisors. Divide the sum of the digits of each of these numbers by 3, and notice the remainder, if any. 4. Make a list of numbers from 1 to 200 which have 9 for one or more of their factors or divisors. Divide the sum of the digits of each by 9, and notice the remainder, if any 97. It is apparent, therefore, that: A number is divisible by 2 if the units' figure is 2, 4, 6, 8, or 0. 3 if the sum of its digits is divisible by 3. 9 if the sum of its digits is divisible by 9. 98. Tell by inspection which of the numbers, on page 129, columns C and D, are divisible by 2; by 5; by 3; by 9. Which in column E are divisible by 2; by 3; by 5; by 9. WRITTEN EXERCISES. 99. 1. What are the prime factors of 336? 2336 2 84 2 42 3 21 3362 x 2 x 2 × 2 × 3 × 7. EXPLANATION.- Since every factor of a number is a divisor of it, we may find the prime factors of 336 by dividing by the exact divisors that are prime numbers. Dividing by 2, we find the factors of 336 to be 2 and 168. But 168 has a factor 2, and, since a factor of a factor of a number is a factor of the number itself, we continue the process, and obtain the prime factors of 336. 100. 1. How many times is 2 times 5 contained in 4 times 5? 2 times 3 in 4 times 3? 2 times any number in 4 times that number? 2. How many times is 4 times 7 contained in 8 times 7? 4 times 35 in 8 times 35? 4 times a certain number in 8 times the same number? 3. How many times is 6 x 12 contained in 18 × 12? 5 x 23 in 15 x 23? 7 x 47 in 21 × 47 ? 4. What is the quotient of (24 × 17) ÷ (12 × 17)? Of (63 × 24) ÷ (9 x 24)? Of (48 × 61) ÷ (24 × 61)? Of (36 times 19) divided by (18 times 19)? 5. In determining the quotient, what numbers may be omitted from both dividend and divisor? 101. From the solution of the examples given, it is evident that: Rejecting equal factors from both dividend and divisor does not alter the quotient. |
